m 


ELEMENTS 

OF 

MACHINE  DESIGN 


''")-    ' 


BY 
O.  A.  LEUTWILER,  M.  E. 

PROFESSOR  OF  MACHINE   DESIGN,   UNIVERSITY   OF  ILLINOIS 

MEMBER  OF  THE  AMERICAN  SOCIETY  OF 

MECHANICAL   ENGINEERS 


T  J 

,  3  6 


FIRST  EDITION 
SECOND  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 

6  &  8  BOUVERIE  ST.,  E.  C. 

1917 


• 


/ 


COPYRIGHT,  1917,  BY  THE 
McGRAw-HiLL  BOOK  COMPANY,  INC. 


THE  MAPLE  PRESS  YORK  PA 


PREFACE 

The  purpose  of  the  author,  in  preparing  this  book,  has  been 
to  present  in  fairly  complete  form  a  discussion  of  the  fundamental 
principles  involved  in  the  design  and  operation  of  machinery.  An 
attempt  is  also  made  to  suggest  or  outline  methods  of  reasoning 
that  may  prove  helpful  in  the  design  of  various  machine  parts. 
The  book  is  primarily  intended  to  be  helpful  in  the  courses  of 
machine  design  as  taught  in  the  American  technical  schools  and 
colleges,  and  it  is  also  hoped  that  it  may  prove  of  service  to  the 
designers  in  engineering  offices. 

Since  a  text  on  machine  design  presupposes  a  knowledge  of 
Strength  of  Materials  and  Mechanics  of  Machinery,  a  chapter 
reviewing  briefly  the  more  important  straining  actions  to  which 
machine  parts  are  subjected  is  included  as  well  as  a  chapter  dis- 
cussing briefly  the  properties  of  the  common  materials  used  in  the 
construction  of  machinery.  Furthermore,  throughout  the  book, 
the  question  of  the  application  of  mechanical  principles  to  ma- 
chines and  devices  has  not  been  overlooked,  and  many  recent 
devices  of  merit  are  illustrated,  described  and  analyzed.  A 
considerable  amount  of  the  material  in  this  book  was  published 
several  years  ago  in  the  form  of  notes  which  served  as  a  text  in  the 
courses  of  machine  design  at  the  University  of  Illinois. 

In  the  preparation  of  the  manuscript  the  author  consulted 
rather  freely  the  standard  works  on  the  subject  of  machine 
design,  the  transactions  of  the  various  national  engineering  socie- 
ties and  the  technical  press  of  America  and  England.  Whenever 
any  material  from  such  sources  of  information  was  used,  the 
author  -endeavored  to  give  suitable  acknowledgment.  The  nu- 
merous illustrations  used  throughout  the  book  have  been  selected 
with  considerable  care  and  in  the  majority  of  cases  they  represent 
correctly  to  scale  the  latest  practice  in  the  design  of  the  parts  of 
modern  machines.  At  the  close  of  nearly  every  chapter  a  brief 
list  of  references  to  sources  of  additional  information  is  given. 

Through  the  generosity  of  various  manufacturers,  drawings 
illustrating  the  prevailing  practice  in  America  were  placed  at  the 
author's  disposal  thus  making  it  possible  to  use  scale  drawings  for 


388062 


vi  PREFACE 

illustrating  the  various  machine  parts.  To  all  such  manufac- 
turers he  is  especially  indebted.  The  author  is  also  indebted  to 
Mr.  H.  W.  Waterfall  of  the  College  of  Engineering  of  the  Univer- 
sity of  Illinois  for  the  many  helpful  suggestions  and  criticisms 
received  during  the  preparation  of  the  manuscript.  To  his  friend 
and  colleague  Professor  G.  A.  Goodenough,  the  author  is  deeply 
indebted  for  much  valuable  advice  and  the  many  suggestions 
received  in  preparing  the  manuscript,  also  for  the  critical  reading 
of  tho  proof. 

O.  A.  L. 
URBANA,  ILLINOIS, 
September,  1917. 


CONTENTS 

CHAPTER  I 

PAQB 

STRESSES  AND  STRAINS  IN  MACHINE  PARTS 1 

Forces — Principles  Governing  Design — Stress  and  Strain — Stress- 
strain  Diagram — Modulus  of  Elasticity — Poisson's  Ratio — Resili- 
ence— Tensile  Stress — Compressive  Stress — Shearing  Stress — 
Stresses  Due  to  Flexure — Flexure  Combined  with  Direct  Stress — 
Straight  Prismatic  Bar — Offset  Connecting  Link — Stresses  in  Col- 
umns— Shearing  Combined  with  Tension  or  Compression — Stresses 
Due  to  Suddenly  Applied  Forces — Repeated  High  Stresses — 
Repeated  Low  Stresses — Safe  Endurance  Stress — Deformation  Due 
to  Temperature  Change — Stresses  Due  to  Temperature  Change — 
Factor  of  Safety — References. 

CHAPTER  II 

MATERIALS  USED  IN  THE  CONSTRUCTION  OF  MACHINE  PARTS 26 

Cast  Iron — Vanadium  Cast  Iron — Pig  Iron — Malleable  Casting — 
Chilled  Casting  —  Semi-steel — Wrought  Iron  —  Manufacturing 
Processes — Manganese-steel  Castings — Applications  of  Manga- 
nese-steel Castings — Bessemer  Process — Open-hearth  Process — 
Cementation  Process — Crucible  Steel — Cold-rolled  Steel — Nickel 
Steel — Chrome  Steel — Vanadium  Steel — Nickel-chromium  Steel — 
Chromium-vanadium  Steel — Silicon-manganese  Steel — Tungsten 
Steel — Brass — Bronze — Monel  Metal — Aluminum — Babbitt  Metal 
— Heat-treating  Processes — S.  A.  E.  Heat  Treatments — Galvanizing 
— Shererdizing. 

CHAPTER  III 

FASTENINGS — RIVETS  AND  RIVETED  JOINTS 48 

Rivets — Rivet  Holes — Forms  of  Rivets — Forms  of  Heads — 
Types  of  Joints — Failure  of  Joints — Definitions — Analysis  of  a 
Boiler  Joint — Efficiency  of  the  Joint — Allowable  Stresses — Mini- 
mum Plate  Thickness — Design  of  a  Boiler  Joint — Rivet  Spacing  in 
Structural  Joints — Types  of  Structural  Joints — Single  Angle  and 
Plate — End  Connections  for  Beams — Double  Angle  and  Plate — 
Splice  Joint — Pin  Plates — Diagonal  Boiler  Brace — References. 

CHAPTER  IV 

FASTENINGS — BOLTS,  NUTS,  AND  SCREWS 76 

Forms  of  Threads— Bolts— Screws— Stay  Bolts— Nut  Locks- 
Washers — Efficiency  of  V  Threads — Stresses  Due  to  Screwing  Up — 
Stresses  Due  to  the  External  Forces — Stresses  Due  to  Combined 
Loads — Fastening  with  Eccentric  Loading — Common  Bearing — 
Efficiency  of  Square  Threads — Stresses  in  Power  Screws. 

vii 


viii  CONTENTS 

CHAPTER  V 

PAGE 

FASTENINGS — KEYS,  COTTERS,  AND  PINS 110 

Sunk  Keys — Keys  on  Flats — Friction  Keys — The  Strength  of  Keys 
— Friction  of  Feather  Keys — Gib-head  Keys — Key  Dimensions — 
Integral  Shaft  Splines — Analysis  of  a  Cotter  Joint — Taper  Pins — 
Rod  and  Yoke  Ends — References. 

CHAPTER  VI 

CYLINDERS  PLATES  AND  SPRINGS 129 

Thin  Cylinders — Thick  Cylinders — Rectangular  Plates — Square 
Plates — Circular  Plates — Flat  Heads  of  Cylinders — Elliptical 
Plates — Helical  Springs — Concentric  Helical  Springs — Helical 
Springs  for  Torsion — Spiral  Springs — Conical  Springs — Leaf 
Springs — Semi-elliptic  Springs — Materials  for  Springs — References. 

CHAPTER  VII 

BELTING  AND  PULLEYS 147 

Leather  Belting— Rubber  Belting— Textile  Belting— Steel  Belting 
— Belt  Fastenings — Tensions  in  Belts — Relation  between  Tight  and 
Loose  Tensions — Coefficient  of  Friction — Maximum  Allowable 
Tension — Selection  of  Belt  Size — Taylor's  Experiments  on  Belting 
— Tandem-belt  Transmission — Tension  Pulleys — Types  of  Pulleys 
— Transmitting  Capacity  of  Pulleys — Proportions  of  Pulleys — 
Tight  and  Loose  Pulleys — Types  of  V  Belts — Force  Analysis  of  V 
Belting — References. 

CHAPTER  VIII 

MANILA  ROPE  TRANSMISSION 17 

Manila  Hoisting  Rope — Sheave  Diameters — Stresses  in  Hoisting 
Rope — Analysis  of  Hoisting  Tackle — Experimental  Data  on 
Hoisting  Tackle — Multiple  System — Continuous  System — Manila 
Transmission  Rope — Sheaves — Relation  between  Tight  and  Loose 
Tensions — Force  Analysis  of  a  Manila  Rope  Transmission — Sheave 
Pressures — Sag  of  Rope — Efficiency  of  Manila  Rope  Drives — 
Selection  of  Rope — Cotton  Rope  Transmission — References. 

CHAPTER  IX 

WIRE  ROPE  TRANSMISSION 195 

Relation  between  Load  and  Effort — Stresses  Due  to  Starting  and 
Stopping — Stresses  Due  to  Bending — Stresses  Due  to  Slack — 
Selection  of  Rope — Hoisting  Tackle — Hoisting  Sheaves  and  Drums 
— Design  of  Crane  Drums — Conical  Drums — Flat  Wire  Ropes — 
Single  Loop  System — Wire  Transmission  Rope — Transmission 
Sheaves — Stresses  in  Wire  Rope — Sag  of  Wire  Rope — References. 


CONTENTS  IX 

CHAPTER  X 

PAGE 

CHAINS  AND  SPROCKETS 216 

Coil  Chain — Stud-link  Chain — Chain  Drums  and  Anchors — Chain 
Sheaves — Relation  between  P  and  Q — Analysis  of  a  Chain  Block 
— Detachable  Chain — Strength  of  Detachable  Chain — Closed-joint 
Chain — Strength  of  Closed-joint  Chain — Sprockets  for  Detachable 
Chain — Relation  between  Driving  and  Driven  Sprockets — Tooth 
Form  —  Rim,  Tooth,  and  Arm  Proportions  —  Block  Chains  — 
Sprockets  for  Block  Chains — Selection  of  Block  Chains — Roller 
Chains — Sprockets  for  Roller  Chains — Length  of  Roller  Chain — 
Silent  Chains — Coventry  Chain — Whitney  Chain — Link  Belt 
Chain — Morse  Chain — Strength  of  Silent  Chain — Sprockets  for 
Silent  Chain — Spring-cushioned  Sprockets — References. 

CHAPTER  XI 

FRICTION  GEARING 259 

Experimental  Results — Plain  Spur  Frictions — Applications  of  Spur 
Frictions — Analysis  of  a  Drop  Hammer — Grooved  Spur  Frictions 
— Starting  Conditions  of  Bevel  Frictions — Running  Conditions — 
Force  Analysis  of  Crown  Frictions — Bearing  Pressures  Due  to 
Crown  Frictions — Friction  Spindle  Press — Curve  Described  by  the 
Flywheel — Pressure  Developed  by  a  Friction  Spindle  Press — 
Double  Crown  Frictions — Efficiency  of  Crown  Friction  Gearing — 
Thrust  Bearing  for  Friction  Gearing — Starting  Loads — References. 

CHAPTER  XII 

SPUR  GEARING 280 

Definitions — Tooth  Curves — Methods  of  Manufacture — Involute 
System — Laying  Out  the  Involute  Tooth — Standard  Involute 
Cutters — Action  of  Involute  Teeth — Cycloidal  System — Form  of 
the  Cycloidal  Tooth — Laying  Out  the  Cycloidal  Tooth — Standard 
Cycloidal  Cutters— Action  of  Cycloidal  Teeth— Strength  of  Cast 
Teeth — Strength  of  Cut  Teeth — Materials  and  Safe  Working 
Stresses — Rawhide  Gears — Fabroil  Gears — Bakelite  Micarta-D 
Gears — Large  Gears — Gear  Wheel  Proportions — Methods  of 
Strengthening  Gear  Teeth — Special  Gears — Efficiency  of  Spur 
Gearing — References. 

CHAPTER  XIII 

BEVEL  GEARING       322 

Methods  of  Manufacture — Form  of  Teeth — Definitions — Acute- 
angle  Bevel  Gears — Obtuse-angle  Bevel  Gears — Right-angle  Bevel 
Gears — General  Assumptions  Regarding  the  Strength  of  Bevel 
Gears— Strength  of  Cast.  Teeth— Strength  of  Cut  Teeth— Method 
of  Procedure  in  Problems — Resultant  Tooth  Pressure — Bearing 


X  CONTENTS 

PAGE 

Pressures  and  Thrusts — Gear  Wheel  Proportions — Non-metallic 
Bevel  Gears — Mounting  Bevel  Gears — Spiral  Bevel  Gears — 
Advantages  and  Disadvantages  of  Spiral  Bevel  Gears — Bearing 
Loads  and  Thrusts  Due  to  Spiral  Bevel  Gears — Experimental 
Results — Skew  Bevel  Gears — References. 

CHAPTER  XIV 

SCREW   GEARING 350 

Types  of  Helical  Gears — Advantages  of  Double-helical  Gears — 
Applications  of  Double-helical  Gears — Tooth  Systems — Strength  of 
Double-helical  Teeth — Materials  for  Helical  Gearing — Double-heli- 
cal Gear  Construction — Mounting  of  Double-helical  Gears — 
Circular  Herringbone  Gears — Straight  Worm  Gearing — Hind- 
ley  Worm  Gearing — Materials  for  Worm  Gearing — Tooth  Forms 
— Load  Capacity — Strength  of  Worm  Gear  Teeth — Force  Analysis 
of  Worm  Gearing — Bearing  Pressures — Worm  and  Gear  Construc- 
tion— Sellers  Worm  and  Rack — Worm  Gear  Mounting. — Tandem 
Worm  Gears — Experimental  Results  on  Worm  Gearing — 
References. 

CHAPTER  XV 

COUPLINGS 383 

Flange  Coupling — Marine  Type  of  Flange  Coupling — Compression 
Coupling — Roller  Coupling — Oldham's  Coupling — Universal  Joint 
— Leather-link  Coupling — Leather-laced  Coupling — Francke  Coup- 
ling— Nuttall  Coupling — Clark  Coupling — Kerr  Coupling — Rolling 
Mill  Coupling — Positive  Clutch — Analysis  of  Jaw  Clutches — 
References. 

CHAPTER  XVI 

FRICTION  CLUTCHES 405 

Requirements  of  a  Friction  Clutch — Materials  for  Contact  Sur- 
faces— Classification  of  Friction  Clutches — Single-cone  Clutch — 
Double-cone  Clutch — Force  Analysis  of  a  Single-cone  Clutch — A 
Study  of  Cone  Clutches — Experimental  Investigations  of  a  Cone 
Clutch — Analysis  of  a  Double-cone  Clutch — Smoothness  of 
Engagement  of  Cone  Clutches — Clutch  Brakes — Single-disc 
Clutch — Hydraulically  Operated  Disc  Clutch — Slip  Coupling — 
Multiple-disc  Clutches — Force  Analysis  of  a  Disc  Clutch — A  Study 
of  Disc  Clutches— Hele-Shaw  Clutch— Ideal  Multi-cone  Clutch- 
Moore  and  White  Clutch — Transmission  Block  Clutches — Analysis 
of  Block  Clutches — Machine-tool  Split-ring  Clutches — Analysis  of 
a  Split-ring  Clutch — Study  of  Split-ring  Clutches — Types  of  Band 
Clutches — Analysis  of  a  Band  Clutch — Horton  Clutch — Require- 
ments of  an  Engaging  Mechanism — Analysis  of  Engaging  Mechan- 
isms— References. 


CONTENTS  xi 

CHAPTER  XVII 

PAGE 

BRAKES 462 

General  Equations — Classification — Single-  and  Double-block 
Brakes — Analysis  of  Block  Brakes — Graphical  Analysis  of  a 
Double-block  Brake — Simple  Band  Brakes — Band  Brakes  for 
Rotation  in  Both  Directions — Differential  Band  Brakes — Conical 
Brakes — Disc  Brakes — Worm-gear  Hoist  Brakes — Crane  Disc 
Brakes — Crane  Coil  Brakes — Cam  Brake — Force  Analysis  of  an 
Automatic  Brake — Disposal  of  Heat — References. 

CHAPTER  XVIII 

SHAFTING 489 

Materials — Method  of  Manufacture — Commercial  Sizes  of  Shaft- 
ing— Simple  Bending — Simple  Twisting — Combined  Twisting  and 
Bending — Method  of  Application — Combined  Twisting  and  Com- 
pression— Bending  Moments — Crane  Drum  Shaft — Shaft  Support- 
ing Two  Normal  Loads  between  the  Bearings — Shaft  Supporting 
Two  Normal  Loads  with  One  Bearing  between  the  Loads — Shaft 
Supporting  One  Normal  and  One  Inclined  Load  between  the  Bear- 
ings— Two-bearing  Shafts  Supporting  Three  Loads — Hollow  Shafts 
— Effect  of  Key-seats  upon  the  Strength  of  Shafts — Effect  of  Key- 
seats  upon  the  Stiffness  of  Shafts — References. 

CHAPTER  XIX 

JOURNALS,  BEARINGS,  AND  LUBRICATION 513 

Types  of  Bearings — General  Considerations — Selection  of  Bearing 
Materials — Provisions  for  Lubrication — Adjustments  for  Wear — 
Adjustments  for  Alignment — Bearing  Pressures — Relation  between 
Length  and  Diameter — Radiating  Capacity  of  Bearings — Coeffi- 
cient of  Friction — Design  Formulas — Temperature  of  Bearings — 
Strength  and  Stiffness  of  Journals — Design  of  Bearing  Caps  and 
Bolts — Work  Lost  Due  to  the  Friction  on  a  Cylindrical  Journal — 
Work  Lost  Due  to  the  Friction  on  a  Conical  Journal — Proportions 
of  Journal  Bearings — Solid  Bearing  with  Thrust  Washers — Collar 
Thrust  Bearings — Step  Bearings — Work  Lost  Due  to  Pivot  Fric- 
tion— Work  Lost  in  a  Collar  Thrust  Bearing — Analysis  of  a  Flat 
Pivot — Tower's  Experiments  on  Thrust  Bearings — Schiele  Pivot 
— References. 

CHAPTER  XX 

BEARINGS  WITH  ROLLING  CONTACT 556 

Requirements  of  Rolling  Contact — Classification — Radial  Bearings 
having  Cylindrical  Rollers — Radial  Bearings  having  Conical 
Rollers — Radial  Bearings  having  Flexible  Rollers — Thrust  Bearing 
having  Cylindrical  Rollers — Thrust  Bearing  having  Conical  Rollers 


xii  CONTENTS 

— Allowable  Bearing  Pressures  and  Coefficients  of  Friction — 
Roller  Bearing  Data — Mounting  of  Roller  Bearings — Forms  of 
Ball  Bearing  Raceways — Experimental  Conclusions  of  Stribeck 
— Radial  Ball  Bearings — Thrust  Ball  Bearings — Combined  Radial 
and  Thrust  Bearing — Allowable  Bearing  Pressures — Coefficient 
of  Friction — Ball  Bearing  Data — Mounting  Ball  Bearings — 
References. 


TABLES 

TABLE  PAGE 

1.  Poisson's  Ratio 6 

2.  Moduli!  of  Resilience  for  Steel  in  Tension 8 

3.  Values  of  Constant  a 21 

4.  Values  of  Coefficient  of  Linear  Expansion 22 

5.  Suggested  Factors  of  Safety 23 

6.  Average  Physical  Properties  of  Principal  Materials 24 

7.  Specifications  of  Pig  Iron 28 

8.  General  Specifications  of  Pig  Iron 29 

9.  Efficiency  of  Boiler  Joints 57 

10.  Ultimate  Shearing  Stresses  in  Rivets 58 

11.  Thickness  of  Shell  and  Dome  Plates  after  Flanging 58 

12.  Thickness  of  Butt  Joint  Cover  Plates 58 

13.  Recommended  Sizes  of  Rivet  Holes 59 

14.  Tension  Members 71 

15.  United  States  Standard  Bolts  and  Nuts 78 

16.  Proportions  of  Sellers  Square  Threads 79 

17.  Proportions  of  Acme  Standard  Threads 80 

18.  Coupling  Bolts 81 

19.  S.  A.  E.  Standard  Bolts  and  Nuts 82 

20.  Standard  Cap  Screws 84 

21.  Standard  Machine  Screws 86 

22.  Safe  Holding  Capacities  of  Set  Screws 87 

23.  Plain  Lock  Washers 94 

24.  Coefficients  of  Friction  for  Square  Threaded  Screws 107 

25.  Bearing  Pressures  on  Power  Screws 108 

26.  Dimensions  of  Woodruff  Keys 112 

27.  Diameters  of  Shafts  and  Suitable  Woodruff  Keys 113 

28.  Round  Keys  and  Taper  Pins 115 

29.  Dimensions  of  Gib-head  Keys 1 19 

30.  S.  A.  E.  Drop  Forged  Rod  and  Yoke  Ends 127 

31.  B.  &  S.  Drop  Forged  Rod  and  Yoke  Ends 128 

32.  Values  of  Coefficients  K,  Ki,  K2,  and  K3 133 

33.  Values  of  Coefficients  K4,  K6,  K6,  and  K7 134 

34.  Values  of  Coefficients  K8  and  K9 136 

35.  Results  of  Test  on  Leather  Belting 149 

36.  Strength  of  Leather  Belt  Joints 155 

37.  Average  Ultimate  Strength  of  Leather  Belting 160 

38.  Working  Stresses  for  Leather  Belting 160 

39.  Comparative  Transmitting  Capacities  of  Pulleys 166 

40.  Proportions  of  Extra-heavy  Cast-iron  Pulleys 167 

41.  Manila  Rope 176 

42.  Hoisting  Tackle  Reefed  with  Manila  Rope 179 

43.  Dimensions  of  Grooves  for  Manila  Rope  Sheaves 184 

xiii 


xiv  TABLES 

TABLE  pAGE 

44.  Dimensions  of  Grooves  for  Manila  Rope  Sheaves 184 

45.  Tensions  due  to  Slack  as  Shown  by  Dynamometer 200 

46.  Steel  Wire  Rope 203 

47.  Hoisting  Tackle  Reefed  with  Wire  Rope 205 

48.  General  Dimensions  of  Wire  Rope  Sheaves 206 

49.  Dimensions  of  Grooves  for  Wire  Rope  Drums 207 

50.  Coefficients  of  Friction  for  Wire  Rope 214 

51.  Hoisting  Chains 217 

52.  Dimensions  of  Grooves  for  Chain  Drums 218 

53.  Dimensions  of  Plain  Chain  Sheaves 221 

54.  Ewart  Detachable  Chain 230 

55.  Closed-joint  Conveyor  and  Power  Chains 232 

56.  Union  Steel  Chains 233 

57.  Sprocket   Teeth  Factors 236 

58.  Diamond  Block  Chain 240 

59.  Diamond  Roller  Chains 244 

60.  Design  Data  for  Morse  Silent  Chain  Drives 252 

61.  Whitney  Silent  Chains  .    .    .    . 253 

62.  Horse  Power  Transmitted  by  Link  Belt  Silent  Chain 254 

63.  General  Proportions  of  Link  Belt  Sprockets 256 

64.  Experimental  Data  Pertaining  to  Friction  Gearing 260 

65.  Radii  for  15°  Involute  Teeth 286 

66.  Brown  and  Sharpe  Standard  Involute  Cutters 287 

67.  Radii  for  Cycloidal  Teeth 290 

68.  Brown  and  Sharpe  Standard  Cycloidal  Cutters 291 

69.  Lewis  Factors  for  Gearing 297 

70.  Lewis  Factors  for  Stub  Teeth 298 

71.  Proportions  of  Cut  Teeth 299 

72.  Values  of  S0  for  Various  Materials 302 

73.  Data  Pertaining  to  Rawhide  Gears 304 

74.  Dimensions  of  Gear  Hubs 310 

75.  Strength  of  Gear  Teeth  used  by  C.  W.  Hunt 312 

76.  Dimensions  of  the  Fellows  Stub  Teeth 313 

77.  Constants  for  Gleason  Unequal  Addendum  Teeth 314 

78.  Experimental  Data  Pertaining  to  Bevel  Gears 349 

79.  Proportions  of  Helical  Teeth 354 

80.  Proportions  of  Fawcus  Double-helical  Teeth 354 

81.  Values  of  Coefficient  K  as  recommended  by  W.  C.  Bates  ....  356 

82.  Cramp's  Gear  Bronzes 367 

83.  Standard  29°  Worm  Threads 368 

84.  Results  of  Tests  on  Cast-iron  Worm  Gearing 381 

85.  Proportions  of  Westinghouse  Flange  Couplings 387 

86.  Dimensions  of  Clamp  Couplings 388 

87.  Dimensions  of  Bocorselski's  Universal  Joints 391 

88.  Dimensions  of  Merchant  &  Evans  Universal  Joints 392 

89.  Data  Pertaining  to  Leather  Link  Couplings 394 

90.  Data  Pertaining  to  Leather  Laced  Couplings 396 

91.  Data  Pertaining  to  Francke  Couplings 398 


TABLES  xv 

TABLE  PAGE 

92.  Service  Factors  for  Francke  Couplings 398 

93.  Proportions  of  Slip  Couplings 431 

94.  Data  Pertaining  to  Various  Types  of  Disc  Clutches 437 

95.  Fiber  Stresses  at  the  Elastic  Limit 499 

96.  Allowable  Bearing  Pressures 528 

97.  Relation  between  Length  and  Diameter  of  Bearings 529 

98.  Dimensions  of  Rigid  Post  Bearings 541 

99.  Coefficients  of  Friction  for  Collar  Thrust  Bearings 552 

100.  Coefficients  of  Friction  for  Step  Bearings 552 

101.  Data  Pertaining  to  Norma  Roller  Bearings 563 

102.  Data  Pertaining  to  Hyatt  High  Duty  Bearings 564 

103.  Crushing  Strength  of  Tool  Steel  Balls 575 

104.  Data  Pertaining  to  Hess-Bright  Radial  Ball  Bearings  .    .        .    .  577 

105.  Data  Pertaining  to  S.  K.  F.  Radial  Ball  Bearings 579 

106.  Data  Pertaining  to  F.  &  S.  Thrust  Ball  Bearings 581 

107.  Data  Pertaining  to  Gurney  Radio-thrust  Ball  Bearings 585 


MACHINE  DESIGN 

CHAPTER  I 
STRESSES  AND  STRAINS  IN  MACHINE  PARTS 

1.  Forces.— The  object  of  a  machine  is  to  transmit  motion 
through  its  various  links  to  some  particular  part  where  useful 
work  is  to  be  done.  The  transmission  of  this  motion  gives  rise 
to  forces  which  must  be  resisted  by  the  parts  of  the  machine 
through  which  the  forces  are  acting. 

The  forces  acting  upon  the  various  machine  parts  may  be 
classified  as  follows: 

(a)  Useful  forces. — In    doing  the  useful  work  for  which  the 
machine  is  intended,  the  various  parts  are  subjected  to  certain 
forces ;  for  example,  the  parts  of  a  shaper  must  transmit  the  forces 
produced  by  the  resistance  offered  to  the  tool  by  the  material  to 
be  cut. 

(b)  Dead  weight  forces. — Dead  weight  forces  are  those  due  to 
the  weights  of  the  individual  parts  in  a  machine.     Generally 
these  forces  are  not  considered  in  the  design  of  a  machine,  except 
in  cranes  and  machines  having  large  gears  and  flywheels.     In  the 
design  of  roof  trusses,  bridges,  structural  steel  towers,  floors,  etc., 
the  dead  weights  are  always  considered,  because  they  form  a  con- 
siderable part  of  the  total  load  coming  upon  the  members. 

(c)  Frictional  forces. — Forces    called  forth  by  the  frictional 
resistances  between  the  machine  parts  are  designated  as  fric- 
tional  forces.     In   certain  classes  of  machinery  such  as  hoists 
employing  screws,  a  large  amount  of  work  is  consumed  by  fric- 
tion; hence  the  various  machine  elements  must  transmit  this 
energy  in  addition  to  energy  required  for  useful  work. 

(d)  Forces  due  to  change  of  velocity. — Frequently  the  motion  of 
machine  parts  changes  in  direction,  thus  causing  forces  that  must 
be  considered ;  for  example,  the  rim  of  a  rapidly  rotating  flywheel 
or  pulley  is  subjected  to  rather  heavy  centrifugal  forces.     Another 
example  is  given  by  the  whipping  action  of  the  connecting  rod  of 
a  high-speed  engine;  the  stresses  arising  from  the  reversal  of 

1 


2  PRINCIPLES  GOVERNING  DESIGN  [CHAP.  I 

direction  may  be  far  in  excess  of  those  due  to  the  steam  pres- 
sure on  the  piston.  In  general,  whenever  the  velocity  of  a 
machine  part  changes  rapidly  heavy  stresses  are  set  up  due  to 
the  accelerating  and  retarding  action.  Forces  due  to  a  change 
of  velocity  are  frequently  called  inertia  forces. 

(e)  Forces  due  to  expansion  and  contraction. — In  certain  struc- 
tures, as  boilers,  the  forces  due  to  the  expansion  and  contraction 
caused  by  heat  must  be  considered.  Not  infrequently  heavy 
bending  stresses  are  induced  in  members  by  expansion  and 
contraction. 

(/)  In  addition  to  the  various  forces  discussed  in  the  preceding 
paragraphs  others  exist,  such  as  the  following:  (1)  forces  due  to 
the  reduction  of  area  caused  by  the  deterioration  of  the  material; 
(2)  force  due  to  the  use  of  non-homogeneous  material;  (3)  forces 
due  to  poor  workmanship. 

Some  of  the  above-mentioned  conditions  need  not  be  considered 
at  all,  but  it  is  well  that  they  all  be  kept  in  mind  when  undertak- 
ing the  design  of  a  new  machine.  It  is  evident  from  this 
brief  discussion,  that  before  the  designer  can  select  a  suitable 
material  or  determine  the  proportions  of  the  various  elements,  he 
must  make  a  careful  analysis  of  the  external  forces  and  their 
effects. 

2.  Principles  Governing  Design. — The  design  of  machine  parts 
may  be  approached  by  either  of  the  following  methods : 

(a)  Strength  alone  is  the  basis  of  design ;  that  is,  the  parts  are 
made  strong  enough  to  resist  the  stresses  developed  in  them,  and 
as  long  as  no  rupture  occurs  the  machine  parts  designed  in  this 
way  fulfill  their  purpose.  As  an  illustration,  the  design  of  a  gear 
transmitting  a  given  horse  power  at  a  certain  speed  is  in  general 
satisfactory  so  long  as  the  various  component  parts  of  the  gear, 
such  as  the  teeth,  rim  and  arms,  do  not  rupture. 

(6)  Stiffness  in  addition  to  strength  is  taken  into  consideration ; 
that  is,  machine  parts  must  be  made  rigid  enough  to  perform  their 
function  without  too  much  distortion.  Stiffness  is  essential  in  the 
design  of  all  the  important  elements  of  a  machine  tool,  as  without 
rigidity  the  machine  is  not  capable  of  producing  work  having  the 
desired  degree  of  accuracy.  A  grinding  machine  is  a  good  illus- 
tration in  the  design  of  which  the  consideration  of  stiffness  is 
paramount. 

The  criterion  discussed  in  (6)  is  important,  and  whenever  pos- 
sible a  study  of  the  deflections  of  the  various  members  of  the  ma- 


ART.  3]  STRESS-STRAIN  DIAGRAM  3 

chine  should  be  attempted.  This  study,  in  the  majority  of  cases, 
is  very  difficult  to  carry  out,  as  the  deflections  are  not  readily 
calculated;  as  a  matter  of  fact,  in  many  cases  it  is  impossible  to 
calculate  such  deflections  with  our  present  knowledge  of  the  sub- 
ject of  "  Strength  of  Materials."  Frequently  the  determina- 
tion of  the  stresses  induced  in  certain  machine  parts  is  beyond 
calculation  and  in  such  cases  as  well  as  those  mentioned  above, 
experience  together  with  precedent  must  be  relied  upon  to  suggest 
the  proper  proportions  to  be  used. 

STRESS,  STRAIN,  AND  ELASTICITY 

3.  Stress  and  Strain.- — The  external  forces  or  loads  coming 
upon  the  members  of  a  machine   cause  the  latter  to  undergo  a 
deformation  or  change  of  form,  the  amount  of  which  is  called  a 
strain.     Now  within  the  member  that  is  thus  deformed,  a  certain 
internal  force  is  produced  in  the  material  which  will  resist  this 
strain.     This  internal  force  is  called  a  stress,  and  may  be  defined 
as  the  internal  resistance  which  the  particles  of  the  material  offer 
to  the  external  force.     A  designer  should  evidently  have  a  knowl- 
edge of  the  stresses  and  strains  induced  in  a  material  subjected  to 
an  external  force,  and  without  such  knowledge  it  is  impossible  for 
him  to  produce  a  well-designed  machine.     Information  pertain- 
ing to  stresses  and  strains  is  derived  from  tests  of  materials.     The 
following  articles  give  briefly  some  of  the  results  of  such  tests. 

4.  Stress-strain  Diagram. — The  relation  existing  between  the 
unit  stresses  and  unit  strains  for  any  particular  material  is  shown 
best  by  means  of  a  diagram.     This  diagram  is  based  upon  the  ob- 
servations and  calculations  derived  from  experiments,  and  is 

•  constructed  by  plotting  upon  rectangular  coordinates  the  unit 
strains  against  the  unit  stresses,  the  latter  as  ordinates  and  the 
former  as  abscissae.  In  Fig.  1  is  shown  such  a  diagram.  The 
plot  shown  represents  the  results  of  a  tensile  test  on  a  soft  grade  of 
machinery  steel.  The  results  of  a  compression  test  on  any  mate- 
rial may  be  plotted  in  a  similar  manner. 

An  inspection  of  the  plot  in  Fig.  1  shows  that  up  to  a  certain 
point  B  the  stress-strain  diagram  is  practically  a  straight  line; 
that  is,  unit  stress  is  proportional  to  unit  strain.  The  law  just 
stated  is  known  as  Hooke's  Law.  The  stress  corresponding  to 
the  point  B  is  known  by  the  term,  "limit  of  proportionality"  or 
better  still  "  proportional  elastic  limit. "  At  the  point  C  there  is  a 


STRESS-STRAIN  DIAGRAM 


[CHAP.  I 


well-defined  break  in  the  diagram,  thus  showing  a  sudden  and  con- 
siderable increase  of  strain  without  an  appreciable  increase  of 
stress;  in  other  words,  this  point  indicates  a  change  in  the  condi- 
tion of  the  material,  namely  from  one  of  almost  perfect  elasticity 
to  one  of  considerable  plasticity.  The  point  C  is  called  the  yield 
point,  and  is  found  only  on  the  stress-strain  diagrams  of  the  duc- 
tile materials.  In  testing  ductile  materials  the  stress  correspond- 
ing to  the  yield  point  is  obtained  by  observing  the  load  on  the 
scale  beam  of  the  machine  at  the  instant  the  beam  takes  a  sudden 
drop. 


50 


•^40 


^30 


—    A 


0  \  F 


0.05 


0.10  0.15 

Unit     Strain 
FIG.  1. 


O.EO 


Another  term  used  considerably  and  frequently  applied  to  the 
stress  corresponding  to  the  point  B  in  Fig.  1  is  the  elastic  limit. 
Various  definitions  have  been  proposed  for  this  term,  and  the 
following  is  considered  about  the  best:  By  the  elastic  limit  is 
meant  the  unit  stress  below  which  the  deformation  or  strain  dis- 
appears completely  upon  removal  of  the  stress ;  in  other  words,  no 
permanent  set  can  be  detected.  The  determination  of  the  elastic 
limit  experimentally  requires  instruments  of  high  precision,  and 
due  to  the  repeated  application  and  release  of  the  stress  that  is 
necessary,  such  tests  require  a  great  amount  of  time.  In  general 
it  is  assumed  that  there  is  but  little  difference  between  the  elastic 
limit  and  the  stress  corresponding  to  the  limit  of  proportionality; 
and  since  the  latter  can  be  determined  more  readily,  it  may  be 


ART.  5] 


MODULUS  OF  ELASTICITY 


used  by  designers  as  a  means  of  getting  at  the  probable  elastic 
limit  of  a  material. 

Referring  again  to  Fig.  1,  it  is  evident  that  as  the  stress  in- 
creases, the  deformation  increases,  until  finally  rupture  of  the 
test  piece  occurs.  The  external  load  required  to  break  the  test 
piece  divided  by  the  original  area  of  cross-section  of  the  bar  is 
called  the  ultimate  strength. 

5.  Modulus  of  Elasticity. — In  order  to  determine  the  strain  for 
any  known  load  acting  upon  a  given  material,  it  is  convenient  to 
make  use  of  the  so-called  modulus  of  elasticity.  This  is  defined  as 


1UU    - 

80 

.Q 
0 

o 

2      60 

c 

(T> 
(I) 

2     40 
+- 
en 

4- 

3      20 
n 

Spring  Steel 

0.  65  C  Steel 
Cold  Rolled 
Medium  Steel 

Soft  Steel 

/ 

t 

/ 

/ 

:=~- 

—  = 

/ 

/_ 
^~ 

^-  —  ' 

= 

/ 

£^ 

* 

— 

^ 

/ 

«^- 

.  — 

/— 

_^ 

/ 

/ 

,  - 

-^- 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

Unit       Strain 
FIG.  2. 

the  ratio  of  unit  stress  to  unit  strain,  a  value  of  which  is  readily 
obtained  from  that  part  of  the  stress-strain  diagram  below  the 
point  J5;  in  other  words,  the  slope  of  the  line  AB  gives  the  value 
of  the  modulus  of  elasticity.  Representing  this  modulus  of  elas- 
ticity for  tension  by  the  symbol  Et)  the  statement  just  made  may 
be  expressed  algebraically  by  the  following  equation : 

Si 

5 


Et  =  ^ 


(1) 


in  which  St  denotes  the  unit  stress  and  5  the  unit  deformation ; 
hence  Et  is  some  quantity  expressed  in  the  same  units  as  St) 
namely  in  pounds  per  square  inch. 


6 


POISSON'S  RATIO 


[CHAP.  I 


In  Fig.  2  are  shown  stress-strain  diagrams  for  several  grades 
of  steel,  which  seem  to  indicate  that  the  modulus  of  elasticity  is 
practically  the  same  for  all  grades  of  steel.  According  to  the 
results  obtained  by  various  authorities  the  numerical  value  of 
the  modulus  of  elasticity  for  steel  varies  from  28,000,000  to  32,- 
000,000  pounds  per  square  inch.  The  modulus  of  elasticity  is  also 
a  measure  of  the  stiffness  or  rigidity  of  a  material,  and  from  Fig.  2 
it  is  evident  that  a  machine  part  made  of  soft  steel  will  be  just 
as  rigid  as  if  it  were  made  of  an  alloy  steel,  provided  the  stresses 
in  the  member  due  to  the  external  load  are  kept  below  the 
limit  of  proportionality.  However,  the  part  when  made  of  high- 
carbon  steel  will  be  much  stronger  than  that  made  of  soft  steel. 
It  has  been  suggested  by  certain  machine-tool  builders  that  ex- 
cessive deflections  of  spindles  and  shafts  may  be  reduced  by  the 
use  of  an  alloy  steel  in  place  of  a  25-point  carbon  open-hearth 
steel,  but  upon  actual  trial  it  was  found  that  the  trouble  was  not 
remedied.  The  failure  of  the  alloy  steel  to  decrease  the  deflec- 
tion, is  due  to  the  fact  that  the  modulus  of  elasticity  and  not  the 
strength  of  the  steel  is  the  measure  of  its  rigidity. 

6.  Poisson's  Ratio. — When  a  bar  is  extended  or  compressed  the 
transverse  dimension  as  well  as  the  length  are  changed  slightly. 

Experimental  data  show  that 
the  ratio  of  the  transverse  unit 
strain  to  the  unit  change  in 
length  is  practically  constant. 
This  ratio  is  called  Poisson's 
ratio,  average  values  of  which, 
collected  from  various  sources, 
are  given  in  Table  1. 

7.  Resilience. — Referring 
to  Fig.  1,  it  is  evident  that  the 
area  under  the  complete  curve  represents  the  work  done  in  ruptur- 
ing the  test  specimen,  while  that  under  the  diagram  up  to  any 
assumed  point  on  the  curve  represents  the  work  done  in  stretch- 
ing the  specimen  an  amount  equal  to  the  deformation  correspond- 
ing to  the  assumed  point.  If  this  assumed  point  be  taken  so 
that  the  stress  corresponding  to  it  is  equal  to  the  elastic  limit, 
then  the  area  under  that  part  of  the  diagram  represents  the  work 
done  in  producing  a  strain  corresponding  to  that  at  the  elastic 
limit.  The  energy  thus  spent  is  called  resilience,  and  is  repre- 


TABLE  1. — POISSON'S  RATIO 


Material 

Poisson's 
ratio 

Cast  iron 

0.270 

Wrought 

iron  ' 

0.278 

Steel  
CoDner 

(Hard 
'  1  Mild 

0.295 
0.303 
0  340 

Brass  .  .  .  . 

0.350 

ART.  7]  RESILIENCE  7 

sented  in  Fig.  1  by  the  triangular  area  AED.     Since  the  area  of 
this  triangle  is  %(AE  X  ED),  it  follows  that 


ASe  ^  SJL      ALSl 
Resilience  =   --  X  --  = 


in  which  A  denotes  the  cross-sectional  area  of  the  test  specimen, 
L  its  length,  and  Se  the  stress  at  the  elastic  limit. 

If  the  specimen  has  a  cross-sectional  area  of  one  square  inch 
and  a  length  of  one  inch,  then  the  second  member  of  (2)  reduces 

sl 

to  x-^r'     This  magnitude  is  then  the  unit  of  resilience   and  is 

^  jiit 

called  the  modulus  of  resilience,  a  quantity  which  is  useful  for 
comparing  the  capacity  various  materials  have  for  resisting  shock. 
As  mentioned  in  Art.  5,  the  modulus  of  elasticity  in  tension  is 
practically  constant  for  the  various  kinds  of  carbon  and  alloy 
steels;  hence  it  follows  from  (2),  that  the  modulii  of  resilience 
of  two  steels  are  to  each  other  as  the  squares  of  the  stresses  at 
their  elastic  limits.  From  this  fact  it  is  apparent  that  the  higher 
carbon  steels  have  a  greater  capacity  for  resisting  shock  than 
those  of  lower  carbon  content,  since  their  elastic  limits  are  higher 
as  shown  in  Fig.  2. 

Unfortunately  writers  on  "  Strength  of  Materials"  have  paid 
but  little  attention  to  the  actual  values  of  the  modulus  of  resilience, 
and  consequently  information  pertaining  thereto  is  not  plentiful. 
The  values  given  in  Table  2  were  calculated  by  means  of  the 
expression  for  the  modulus,  and  may  serve  as  a  guide  in  the  proper 
selection  of  a  shock-resisting  material.  The  stresses  at  the 
elastic  limit  given  in  Table  2  were  collected  from  various  sources, 
and  in  all  probability  in  the  majority  of  cases,  the  yield  point 
instead  of  the  elastic  limit  is  referred  to.  The  error  introduced 
by  substituting  the  stress  at  the  yield  point  for  that  at  the 
elastic  limit  is  not  of  great  consequence  since  the  values  given 
in  Table  2  serve  merely  as  a  guide. 

It  should  be  noted  that  the  preceding  discussion  of  resilience 
applies  to  the  stress-strain  diagram  given  in  Fig.  1  which  repre- 
sents the  result  of  a  tensile  test;  however,  the  formula  for  the 
modulus  of  resilience  applies  also  to  direct  compressive  or  shear- 
ing stresses,  provided  the  modulus  of  elasticity  and  the  stress  at 
the  elastic  limit  are  given  their  appropriate  values. 


TENSILE  STRESS  [CHAP.  I 

TABLE  2. — MODULI  OF  RESILIENCE  FOB  STEEL  IN  TENSION 


Type  of  steel 

Elastic 
limit 

Modulus  of 
resilience 

Open-hearth 
carbon  steels 

0.08  per  cent.  C. 

25,000 
30,000 
35,000 
41,000 
47,500 
63,500 
70,500 
75,000 

10.4 
15.0 
20.4 
28.0 
37.6 
67.2 
82.8 
93.8 

0.15  per  cent  C 

0.30  per  cent.  C.   .  . 

0.40  per  cent.  C  
0.50  per  cent   C. 

0.60  per  cent.  C  

0.70  per  cent.  C. 

0.80  per  cent  C 

Alloy  steels 

Nickel  steel,  2.85  per 
cent.  Ni. 

Annealed 

52,000 

45.1 

Oil-tempered 

121,000 

244.0 

Chrome  steel,  oil-tempered  

127,500 

271.0 

Carbon  vanadium,  oil-tempered.  .  .  . 

136,000 

308.0 

Nickel  vanadium,  oil-tempered  

126,250 

266.0 

Chrome  vanadium.  .  . 

Annealed 

63,700 

67.5 

Oil-tempered 

170,000 

482.0 

SIMPLE  STRESSES 

The  external  forces  acting  upon  a  machine  part  induce  various 
kinds  of  stresses  in  the  material,  depending  upon  the  nature  of 
these  forces.  The  different  kinds  of  stresses  with  which  a  designer 
of  machines  comes  into  contact  will  now  be  discussed  briefly. 

8.  Tensile  Stress. — A  machine  member  is  subjected  to  a  ten- 
sile stress  when  the  external  forces  acting  upon  it  tend  to  pull  it 
apart.  Using  the  notation  given  below,  the  relations  existing  be- 
tween stress,  strain  and  the  external  forces  for  the  case  of  simple 
tension  are  derived  as  follows: 

Let  A  =  cross-sectional  area  of  the  member, 
Et  =  modulus  of  elasticity. 
L  =  length  of  the  member 
P  =  the  external  force. 
St  =  unit  tensile  stress. 
A  =  total  elongation. 


ART.  9]  SHEARING  STRESS  9 

The  area  of  cross-section  of  the  member  multiplied  by  the 
unit  stress  gives  the  total  stress  induced  in  the  section,  and  since 
the  total  stress  induced  is  that  due  to  the  pull  of  the  force  P, 
it  follows  that 

8.  =  |  (3) 

From  the  definition  of  the  modulus  of  elasticity  given  in  Art. 
5,  or  from  (1),  we  get 

Et  =  ^,  (4) 

from  which  the  following  expression  for  the  total  elongation  is 
obtained  : 


By  means  of  (5)  ,  it  is  possible  to  determine  the  probable  elonga- 
tion of  a  given  member  subjected  to  a  load  P.  This  is  a  very 
desirable  thing  to  do  for  all  tension  members  of  considerable 
length,  as  very  frequently  such  elongation  is  limited  by  the  class 
of  service  for  which  the  proposed  machine  is  intended. 

9.  Compressive  Stress.  —  A  compressive  stress  is  induced  in  a 
member  when  the  external  forces  tend  to  force  the  particles  of  the 
material  together.     For  a  short  member,  in  which  no  buckling 
action  is  set  up  by  the  external  forces,  the  various  relations  de- 
duced in  Art.  8  apply  also  in  this  case,  provided  the  appropriate 
values  are  substituted  for  the  various  symbols.     If,  however,  the 
length  of  the  member  exceeds  say  six  times  the  least  diameter, 
the  stresses  induced  must  be  determined  by  the  column  formulas, 
which  are  discussed  in  Art.  15. 

A  kind  of  compressive  stress  met  with  extensively  in  designing 
machinery  is  that  caused  by  two  surfaces  bearing  against  each 
other;  for  example,  the  edges  of  plates  against  rivets  or  pins,  or 
keys  against  the  sides  of  the  key-  way  or  key-seat.  This  kind  of 
a  stress  is  usually  spoken  of  as  a  bearing  stress. 

10.  Shearing  Stress.  —  A  shearing  stress  is  one  that  is  produced 
by  the  action  of  external  forces  whose  lines  of  action  are  parallel 
and  in  opposite  direction  to  each  other.     The  relation  existing 
between  the  external  force  P,  area  of  cross-section  A,  and  the 
shearing  stress  S8,  is  similar  to  (3),  or 

&  =  (6) 


10  SHEARING  STRESS  [CHAP.  1 

If  a  machine  member  is  twisted  by  a  couple,  the  stress  induced 
in  that  member  is  a  pure  shear,  or  as  it  is  commonly  called,  a 
torsional  stress.  The  following  discussion  establishes  the  relations 
existing  between  stress,  strain  and  the  external  forces  for  a 
member  having  a  circular  cross-section. 

Equating  the  external  moment  T  to  the  internal  resisting 
moment,  we  obtain 

T  =  ^,  (7) 

in  which  J  represents  the  polar  moment  of  inertia  and  d  the  di- 
ameter of  the  member.  For  any  given  section  the  value  of  J 
may  be  obtained  by  means  of  the  relation: 

J  =  Ii  +  /2,  (8) 

in  which  /i  and  /2  represent  the  rectangular  moments  of  inertia 
of  the  section  about  any  two  axes  at  right  angles  to  each  other, 
through  the  center  of  gravity.  For  a  circular  cross-section 

ird* 
J  =  2  Ii  =  -^,  hence  (7)  becomes 

T  =  ^  (9) 

The  relation  between  the  twisting  moment  T  and  the  angular 
deflection  6  of  a  circular  member  having  a  length  L  is  derived  in 
the  following  manner: 


Substituting  in  (9)  the  value  of  Ss  from  (10),  we  obtain 

T  -  -^  (11) 

584  L 

The  expression  given  by  (9)  is  to  be  used  when  the  member 
must  be  designed  for  strength,  while  (11)  is  used  to  proportion 
the  member  for  stiffness. 

11.  Stresses  Due  to  Flexure.  —  Machine  members  may  be 
subjected  to  transverse  forces  which  produce  stresses  of  several 
kinds.  Such  members  must  be  designed  by  considering  the 
effect  produced  by  the  combination  of  these  several  stresses.  A 
simple  illustration  of  a  member  in  which  several  kinds  of  stresses 
are  induced  is  an  ordinary  beam  supported  at  its  ends  and  carry- 
ing a  load  W  at  a  distance  x  from  the  left-hand  support.  Due 


ART.  12]  STRESSES  DUE  TO  FLEXURE  11 

to  the  load  W ,  the  beam  will  bend  downward  producing  a  com- 
pressive stress  on  the  upper  or  concave  side,  a  tensile  stress  on  the 
lower  or  convex  side,  and  a  shearing  stress  at  right  angles  to  the 
tensile  and  compressive  stresses  just  mentioned.  In  calculations 
pertaining  to  beams,  the  magnitude  of  the  shearing  stress  is 
generally  small  relative  to  the  tensile  and  compressive  stresses, 
and  may  then  be  neglected  altogether;  however,  cases  may  arise 
when  the  shearing  stress  must  be  considered. 

The  relation  existing  between  the  bending  moment  produced 
in  the  beam  by  the  load  W,  the  stress  S  and  the  dimensions  of  the 
cross-section  of  the  beam,  is  obtained  by  equating  the  external 
moment  to  the  internal  stress  moment;  thus 

M  =  — ,  (12) 

c 

in  which  /  represents  the  moment  of  inertia  of  the  beam's  cross- 
section,  and  c  the  distance  from  the  center  of  gravity  of  the  sec- 
tion to  the  outermost  fiber.  This  formula  is  applicable  for  de- 
termining the  strength  of  the  beam,  provided  S  is  kept  within 
the  elastic  limit. 

Whenever  a  beam  is  to  be  designed  for  stiffness  the  following 
general  formula  may  be  used : 

M  =  El  g  (13) 

The  expression  given  by  (13)  is  the  fundamental  equation  by 
means  of  which  the  deflection  of  any  beam  may  be  obtained. 
The  method  of  procedure  is  to  determine,  for  the  case  considered, 
an  expression  for  the  bending  moment  M  in  terms  of  x,  and  after 
substituting  it  in  (13),  integrate  twice  and  solve  for  the  vertical 
deflection  y  of  the  beam. 

COMBINED  STRESSES 

12.  Flexure  Combined  with  Direct  Stress. — In  structures  such 
as  bridges  and  roofs,  the  members  are,  in  general,  pieces  that  are 
acted  upon  by  equal  and  opposite  forces.  There  being  no  motion 
at  the  joints,  it  is  properly  assumed  that  such  members  are  cen- 
trally loaded,  thus  producing  a  uniformly  distributed  stress  in 
the  material.  When,  however,  we  deal  with  machine  parts, 
central  loading  is  the  exception  rather  than  the  rule.  Even  in 
the  ordinary  connecting  link  used  merely  to  transmit  motion, 


12 


STRAIGHT  PRISMATIC  BAR 


[CHAP.  I 


the  friction  between  the  pin  and  its  bearing  in  the  link  causes  a 
shifting  of  the  line  of  action  by  an  appreciable  amount,  thus 
subjecting  the  link  to  a  flexural  stress  in  addition  to  the  direct 
stress. 

In  order  to  determine  the  distribution  of  stress  in  any  right 
section  of  a  member  subjected  to  flexure  combined  with  direct 
stress,  and  thence  to  find  the  maximum  intensity  of  stress,  the 
following  analysis  and  discussion  is  recommended.  Attention  is 
called  to  the  fact  that  the  expressions  given  are  strictly  applicable 
only  to  the  following  types  of  members  : 

(a)  Short  as  well  as  long  tension  members  that  are  straight. 

(6)  Short  and  straight  compression  members. 

13.  Straight  Prismatic  Bar.  —  In  Fig.  3  is  shown  a  straight  pris- 
matic bar  so  loaded  that  the  line  of  action  of  the  external  force 
P  is  parallel  to  the  axis  AB  and  at  a  dis- 
tance e  from  it.  We  are  to  determine  the 
distribution  of  stress  in  any  right  section 
as  CD  and  thence  to  find  the  maximum  in- 
tensity of  stress.  Consider  the  portion  of 
the  bar  above  CD  as  a  free  body,  and  at 
the  center  0  of  the  section  insert  two  op- 
posite forces  OM  and  ON  acting  along 
the  axis  AB,  and  equal  to  the  external 
force  P.  These  forces  being  equal  and 
opposite,  do  not  affect  the  equilibrium 
of  the  system.  We  have  thus  replaced 
the  single  external  force  by  the  central 
force  OM  and  a  couple  consisting  of  the 
equal  and  opposite  forces  P  and  ON.  The 
arm  of  the  couple  is  e  and  its  moment  is 
Pe.  The  single  force  OM  must  be 
balanced  by  a  stress  in  the  section  CD; 
and  since  OM  has  the  axis  A  B  as  its  line  of  action,  this  stress 
is  uniformly  distributed  over  the  cross-section.  Denoting  the 
intensity  of  this  stress  by  S't,  and  the  area  of  the  section  by  A, 
we  have 


P 

A 

^ft— 

-  c  —  • 

-»€ 

j- 

G 

N 

k' 

M 

c- 

0 

1 

-D 

N 

i 

\ 

N 

\ 
\ 

H 

B 

P 

FIG.  3. 


S't  =- 


(14) 


If  the  intensity  S't,  be  denoted  in  Fig.  3  by  CE,  the  line  EF 
parallel  to  CD  will  indicate  graphically  the  uniform  distribution 
of  stress  over  the  section. 


ART.  13]  STRAIGHT  PRISMATIC  BAR  13 

The  couple  of  moment  Pe  tends  to  give  the  body  under  con- 
sideration a  counter-clockwise  rotation.  Evidently  this  couple 
must  be  balanced  by  a  stress  with  an  equal  moment  and  of 
opposite  sense.  The  fibers  to  the  right  of  AB  will  be  subjected 
to  tensile  stress  and  those  to  the  left  to  compressive  stress.  De- 
noting the  intensity  of  the  flexural  stress  at  D  by  S",  and  the 

section  modulus  of  the  section  by  —  ,  then 


Ct 

Pect 


,1F.N 
(15) 


Denoting   the  intensity  of   the  flexural  stress  at  the  point    C 
by  S",  and  the  section  modulus  of  the  section  by  —  ,  we  get 


S     =  (16) 

The  law  of  distribution  of  the  stress  induced  by  the  couple  Pe 
is  represented  graphically  by  the  line  GH.  Evidently  the  maxi- 
mum intensity  of  tensile  stress  occurs  at  the  point  D,  and  its 
magnitude  is  obtained  by  adding  (14)  and  (15),  or 


=  T-      1  +  -^P  (IV) 


The  maximum  compression  stress  occurs  at  the  point  C,  and 
its  magnitude  is  given  by  the  following  expression: 


Sc  =  -  l  (18) 

Equations  (17)  and  (18)  are  not  strictly  exact,  since  the  flexural 
stresses  S"  and  S"  do  not  represent  actual  direct  stresses  and 
therefore  should  not  be  combined  directly  with  the  true  direct 
stress  St.  The  difference  between  these  stresses  may  be  con- 
siderable for  materials  in  which  the.  rates  of  deformation  due  to 
tension  and  compression  are  not  equal,  as  in  cast  iron,  brass,  and 
wood.  A  better  method  would  be  to  express  S't'  and  S"  in 
terms  of  S't  before  combining  them  with  the  latter.  In  'general, 
to  express  a  stress  due  to  flexure  in  terms  of  a  direct  stress,  multi- 
ply the  former  by  the  ratio  that  the  direct  stress  of  the  given 
material  bears  to  the  transverse  stress. 

In  the  analysis  just  given  the  external  force  P  produces  a  direct 
tensile  stress  over  the  area  A;  however,  the  various  formulas 
derived  above  apply  to  the  condition  when  the  force  P  is  reversed, 


14 


OFFSET  CONNECTING  LINK 


[CHAP.  I 


namely  producing  a  direct  compressive  stress,  providing  the 
proper  symbols  are  used. 

14.  Offset  Connecting  Link.  —  A  case  of  frequent  occurrence 
in  the  design  of  machine  parts  is  the  offset  connecting  link  shown 
in  Fig.  4.     The  circumstances  are  such  that  it  is  not  practicable 
to  make  the  link  straight,  and  the  axis  of  a  cross-section,  as  CD, 
lies  at  a  distance  e  from  the  line  AB, 
which  joins  the  centers  of  the  pins  and  is, 
therefore,  neglecting  friction,  the  line  of 
action  of  the  external  forces.     Let  fro  and 
ho  denote  the  dimensions  of  the  rectan- 
gular cross-section  of  the  link  if  straight 
and  centrally  loaded;  and  let  b  and  h  de- 
note the  corresponding  dimensions  of  the 
eccentrically  loaded  section  at  CD. 

For  the  straight  link  the  intensity  of 
the  uniformly  distributed  stress  is 


So  = 


oio 


(19) 


For  the  offset  link  the  maximum  intensity 
of  stress  in  the  section  CD  as  calculated 
by  means  of  (17)  is 
FIG.  4.  P  r6  e 

If  we  impose  the  condition  that  St  shall  not  exceed  So,  we  have 

bh  ^  b0hQ  l~  +  l]  (21) 

L  h          J 

Let  mho  denote  the  distance  of  the  right-hand  edge  of  the  cross- 
section  CD  from  AB,  the  line  of  action  of  the  external  forces; 
this  is  to  be  taken  positive  when  measured  from  AB  to  the  right, 
that  is,  when  AB  cuts  the  section  in  question,  and  negative  when 
measured  from  AB  to  the  left.  Then  the  eccentricity  is 


e  =  -  —  mho 

Substituting  this  value  in  (21),  we  have  finally 

, ,   .    ,  ,    r,       6ra/i0~| 
bh  ^  b0ho  14 j— 


(22) 


(23) 


ART.  15]  STRESSES  IN  COLUMNS  15 

A  discussion  of  (23)  leads  to  some  interesting  results.  For 
given  values  of  60/to  and  m,  we  may  vary  b  and  h  as  we  choose, 
subject  to  the  restriction  expressed  by  (23).  Economy  of 
material  is  obtained  by  making  the  product  bh  and,  therefore, 

the  expression     4  --  T—      as  small  as  possible.     If  m  is  posi- 

tive, that  is,  if  the  seetion  is  cut  by  the  line  of  action  of  the 
forces,  this  requirement  is  met  by  making  h  as  small  as  possible; 
on  the  other  hand,  if  m  is  negative,  that  is,  if  the  section  CD  lies 
wholly  outside  of  the  line  of  action  of  P,  the  product  bh  is  made 
a  minimum  by  making  h  as  large  as  possible.  In  other  words, 
when  m  is  positive,  keep  the  width  h  as  small  as  possible  and 
increase  the  area  of  the  section  by  increasing  the  thickness  6; 
when  m  is  negative,  keep  the  thickness  b  small  and  add  to  the 
area  of  the  section  by  increasing  the  width  h.  This  principle 
is  of  importance  in  the  design  of  the  C-shaped  frames  of  punches, 
shears,  presses  and  riveters. 

When  m  =  0,  that  is  when  the  edge  of  the  section  coincides  with 
the  line  of  action  AB,  (23)  reduces  to  bh  ^  4  b0hQ.  The  area  of 
section  bh  must  be  at  least  four  times  the  area  of  section  b0ho, 
independent  of  the  relative  dimensions  of  the  section. 

15.  Stresses  in  Columns.  —  As  stated  in  Art.  9,  the  formulas  for 
short  compression  members  are  not  applicable  to  centrally  loaded 
compression  members  whose  length  is  more  than  six  times  its 
least  diameter.  Due  to  the  action  of  the  external  load,  such  a 
member  will  deflect  laterally,  thus  inducing  bending  stresses 
in  addition  to  the  direct  stress. 

(a)  Hitter's  formula.  —  Many  formulas  have  been  proposed 
for  determining  the  permissible  working  stress  in  a  column  of 
given  dimensions.  Some  of  these  are  based  upon  the  results  ob- 
tained from  tests  on  actual  columns,  while  others  are  based  on 
theory.  In  1873,  Hitter  proposed  a  rational  formula,  by  means  of 
which  the  value  for  the  mean  intensity  of  permissible  compressive 
stress  in  a  long  column  could  be  determined.  This  formula, 
given  by  (24)  is  used  generally  by  designers  of  machine  parts: 


(24) 


in  which 

A  =  area  of  cross-section. 
E  =  coefficient  of  elasticity. 


16  STRESSES  IN  COLUMNS  [CHAP.  I 

L  =  the  unbraced  length  of  the  column  in  inches. 

P  =  the  external  load  on  the  column. 

Sc  =  the  greatest  compressive  stress  on  the  concave  side. 

Se  =  unit  stress  at  the  elastic  limit. 

n  =  a  constant. 

r  =  least  radius  of  gyration  of  the  cross-section. 

The  strength  of  a  column  is  affected  by  the  condition  of  the 
ends,  that  is  the  method  of  supporting  and  holding  the  columns. 
In  (24)  this  fact  is  taken  care  of  by  the  factor  n,  which  may  have 
the  following  values,  taken  from  Merriman's  "  Mechanics  of 
Materials." 

1.  For  a  column  fixed  at  one  end  and  free  at  the  other,  n  =  0.25. 

2.  For  a  column  having  both  ends  free  but' guided,     n  =  1. 

3.  For  a  column  having  one  end  fixed  and  the  other  guided, 

n  =  2.25. 

4.  For  a  column  having  both  ends  fixed  n  =  4. 
(b)  Straight  line  formula. — A  formula  used  very  extensively 

by  structural  designers  is  that  proposed  by  Mr.  Thos.  H.  Johnson, 
and  is  known  as  the  straight  line  formula.  It  is  not  a  rational 
formula,  but  is  based  on  the  results  of  tests.  Using  the  same 
notation  as  in  the  preceding  article,  Johnson's  straight  line 
formula  for  the  mean  intensity  of  permissible  compressive  stress  is 

S'°  =  Z  =  Sc  ~  T'  (25) 

in  which  C  is  a  coefficient  whose  value  may  be  determined  by  the 
following  expression: 

0          / A    cr 

(26) 

The  factor  n  in  (26)  has  the  same  values  as  those  used  in  con- 
nection with  Hitter's  formula  given  above. 

The  straight  line  formula  has  no  advantage  over  the  Hitter 
formula  as  far  as  simplicity  is  concerned,  except  possibly  in  a 
series  of  calculations  in  which  the  value  of  C  remains  constant, 
as,  for  example,  in  designing  the  compression  members  of  roof 
trusses  in  which  the  same  material  is  used  throughout.  For  a 
more  complete  discussion  of  the  above  formulas  the  reader  is 
referred  to  Mr.  Johnson's  paper  which  appeared  in  the  Transac- 
tions of  the  American  Society  of  Civil  Engineers  for  July,  1886. 


ART.  17]  COMBINED  STRESSES  17 

16.  Eccentric  Loading  of  Columns.  —  Not  infrequently  a  de- 
signer is  called  upon  to  design  a  column  in  which  the  external 
force  P  is  applied  to  one  side  of  the  gravity  axis  of  the  column;  in 
other  words,  the  column  is  loaded  eccentrically.  A  common 
method  in  use  for  calculating  the  stresses  in  such  a  column 
consists  of  adding  together  the  following  stresses: 

(a)  The  stress  due  to  the  column  action  as  determined  by 

Dp  O    T  2    -i 

means  of  the  Ritter  formula,  or  -T-    1  H  —  *  2  2   • 

Pec 

(b)  The  flexural  stress   due  to  the  eccentricity,  namely 


in  which  c  is  the  distance  from  the  gravity  axis  of  the  column  to 
the  outer  fiber  on  the  concave  side,  and  e  is  the  eccentricity  of  the 
external  force  P,  including  the  deflection  of  the  column  due  to  the 
load.  For  working  stresses  used  in  designing  machine  members, 

the  deflections  of  columns  having  a  slenderness  ratio  —  of  less 

than  120  are  of  little  consequence  and  for  that  reason  may  be 
neglected,  thus  simplifying  the  calculations. 

By  adding  the  two  stresses  we  find  that  the  expression  for  the 
maximum  compressive  stress  in  an  eccentrically  loaded  column  is 


17.  Shearing  Combined  with  Tension  or  Compression.  —  Many 
machine  members  are  acted  upon  by  external  forces  that  produce 
a  direct  tensile  or  compressive  stress  in  addition  to  a  direct  shear- 
ing stress'  at  right  angles  to  the  former.  The  combination  of 
these  direct  stresses  produces  similar  stresses,  the  magnitudes 
of  which  may  be  arrived  at  by  the  following  expressions  taken 
from  Merriman's  "  Mechanics  of  Materials:" 


S  /  S 2 

Maximum  tensile  stress  =  -=-'  +  \/&2  +  -7-  (28) 

Cf  / a  2 

Maximum  compressive  stress  =  -^  +  \JS32  +  -j-  (29) 

/ o  2  / Q  2 

Maximum  shearing  stress         =  -\ISa2  +  -j-  or  \/^»2  +  ~T~ 

These  formulas  will  be  found  useful  in  arriving  at  the  resultant 
stresses  in  machine  members  subjected  to  torsion  combined  with 
bending  or  direct  compression.  Such  cases  will  be  discussed  in 
the  chapter  on  shafting. 


18 


SUDDENLY  APPLIED  STRESSES 


[CHAP.  I 


18.  Stresses  Due  to  Suddenly  Applied  Forces. — In  studying 
the  stresses  produced  by  suddenly  applied  forces,  two  distinct 
cases  must  be  considered. 

(a)  An  unstrained  member  acted  upon  by  a  suddenly  applied 
force  having  no  velocity  of  approach. 

(6)  An  unstrained  member  acted  upon  by  a  force  that  has  a 
velocity  of  approach. 

CASE  (a) . — For  the  case  in  which  the  suddenly  applied  force  P 
has  no  velocity  before  striking  the  unstrained  member,  the  exter- 
nal work  done  by  this  force  is  PA,  in  which  A 
represents  the  total  deformation  of  the  mem- 
ber. If  the  stress  S  induced  in  the  member 
having  an  area  A  does  not  exceed  the  elastic 
limit,  then  the  internal  work  is  represented 
by  the  following  expression: 


Internal  work 


AAS 


Equating  the  external  to  the  internal  work, 
we  obtain 

9  P 

S  =  2-f  (31) 

That  is,  the  stress  produced  in  this  case  by 
the  suddenly  applied  force  P  is  double  that 
produced  by  the  same  force  if  it  were  applied 
gradually. 

CASE  (6) . — To  derive  the  expression  for  the 
magnitude  of  the  stress  induced  in  an  un- 
strained member  of  area  A  by  a  force  P  that 
has  a  velocity  of  approach  v,  we  shall  assume  a  long  bolt  or  bar 
having  a  head  at  one  end  and  the  other  end  held  rigidly  as 
shown  in  Fig.  5.  Upon  the  bolt  a  weight  W  slides  freely,  and 
is  allowed  to  fall  through  a  distance  b  before  it  strikes  the  head 
of  the  bolt. 

As  soon  as  the  weight  W  strikes  the  head,  the  bolt  will  elongate 
a  distance  A,  from  which  it  is  evident  that  the  external  work 
performed  by  W  is  W(b  +  A).  The  stress  in  the  bolt  at  the 
instant  before  W  strikes  the  head  is  zero,  and  after  the  bolt  has 
been  elongated  a  distance  A  the  stress  is  S;  hence  the  work  of  the 

variable  tension  during  the  period  of  elongation  is  ~ 7T~>  assuming 


ART.  19]  REPEATED  STRESSES  19 

that  S  is  within  the  elastic  limit.  To  do  this  internal  work,  the 
weight  W  has  given  up  its  energy;  hence  equating  the  external 
to  the  internal  work  and  solving  for  S,  we  get 


+  A)  (32) 

From  Art.  5,  the  elongation 

SL 
A  =  lf 

Substituting  this  value  of  A  in  (32),  and  collecting  terms 


If  in  (33),  the  distance  b  is  made  zero,  so  as  to  give  the  condi- 

2  W 

tions  stated  in  case  (a)  above,  we  find  that  S  =  ~r~,  which  agrees 

with  results  expressed  by  (31). 

REPEATED  STRESSES 

19.  Repeated  High  Stresses.  —  It  is  now  generally  conceded 
that  in  a  machine  part  subjected  to  repeated  stress  there  is  some 
internal  wear  or  structural  damage  of  the  material  which  eventu- 
ally causes  failure  of  the  part.  In  June,  1915,  Messrs.  Moore 
and  Seely  presented  before  the  American  Society  for  Testing 
Materials  a  paper,  in  which  they  gave  an  excellent  analytical 
discussion  of  the  cumulative  damage  done  by  repeated  stress. 
The  application  of  the  proposed  formula  gives  results  that  agree 
very  closely  with  the  experimental  results  obtained  by  the  authors 
themselves  as  well  as  those  obtained  by  earlier  investigators.  For 
a  range  of  stress  extending  from  the  yield  point  to  a  stress  slightly 
below  the  elastic  limit,  Messrs.  Moore  and  Seely  derived  the 
following  formula  as  representing  the  relation  existing  between 
the  fiber  stress  and  the  number  of  repetitions  of  stress  necessary 
to  cause  failure: 

S  =  (T^V  (34) 

in  which 

N  =  the  number  of  repetitions  of  stress. 

S  =  maximum  applied  unit  stress  (endurance  strength). 

a  =  constant  depending  upon  the  material. 


20 


REPEATED  STRESSES 


[CHAP.  I 


b  =  constant  based  upon  experiment, 
minimum  unit  stress 
maximum  unit  stress 

For  a  complete  reversal  of  stress,  q  =  —  1,  and  when  the  range 
is  from  zero  to  a  maximum,  q  =  0. 

20.  Repeated  Low  Stresses. — The  formula  expressing  the  re- 
lation between  the  fiber  stress  and  the  number  of  repetitions  of 
low  stress,  according  to  the  above-mentioned  paper,  is  as  follows : 


S  = 


(1  -  q)N" 


+cNe), 


(35) 


in  which  c  and  e  are  constants,  the  values  of  which  must  be  ob- 
tained by  means  of  experiments.     The  factor  (1  +  cNe)  is  called 


Number     of     Repetitions 


1.30 


0      I0e 


5xl06 


10 '  5xl07 

Number    of     Repetitions 
FIG.  6. 


by  the  authors  a  probability  factor,  and  its  numerical  value  de- 
pends altogether  upon  the  judgment  of  the  designer.  In  Fig.  6 
are  plotted  the  values  of  (1  +  cNe),  as  proposed  by  the  authors, 
for  use  in  determining  the  magnitude  of  the  stress  S  in  any  part, 
the  failure  of  which  would  not  endanger  life.  For  parts,  the  fail- 
ure of  which  would  endanger  life,  this  probability  factor  should 
be  assumed  as  equal  to  unity.  In  Table  3  are  given  values  of 
a  for  various  materials,  as  determined  from  existing  data  of 
repeated  stress  tests. 


ART.  21]  SAFE  ENDURANCE  STRESS 

TABLE    3. — VALUES   OF   CONSTANT   a 


21 


Material 

a 

Material 

a 

Structural  steel. 

250,000 

Spring  steel 

400  000  to 

Soft  machinery  steel  ....... 
Cold-rolled  steel  shafting  .  .  . 

250,000 
400,000 

Hard-steel  wire  .... 
Gray  cast  iron. 

600,000 
600,000 
100  000 

Steel  (0.45  per  cent,  carbon) 
Wrought  iron     

350,000 
250,000 

Cast  aluminum  .... 
Hard-drawn  copper 

80,000 
140  000 

wire. 

The  value  of  q,  the  ratio  of  minimum  to  maximum  stress  is 
usually  known,  or  may  be  established  from  the  given  data.  Ac- 
cording to  the  authors,  if  the  stress  is  wholly  or  partially  reversed, 
q  must  be  taken  as  negative,  having  a  value  of  —  1  when  there 
is  a  complete  reversal  of  stress.  In  cases  where  the  value  of  q 
approaches  +1,  it  is  possible  that  the  endurance  stress  calcu- 
lated by  means  of  (35),  will  be  in  excess  of  the  safe  static  stress,  in 
which  case  the  latter  should  govern  the  design. 

For  the  exponent  6,  Messrs.  Moore  and  Seely  recommend  that 
it  should  be  made  equal  to  %,  this  value  being  derived  from  a 
careful  study  of  data  covering  a  wide  range  of  repeated  stress 
tests. 

21.  Safe  Endurance  Stress. — As  stated  in  a  preceding  para- 
graph, the  formula  given  applies  only  to  stresses  up  to  the  yield 
point  of  the  material;  hence  whenever  the  endurance  strength 
calculated  by  (35)  is  less  than  the  yield  point,  a  so-called  factor 
of  safety  must  be  introduced,  in  order  to  arrive  at  a  safe  endurance 
stress.  This  may  be  accomplished  in  the  following  two  ways: 

(a)  By  applying  the  factor  of  safety  to  the  stress. 

(6)  By  applying  the  factor  of  safety  to  the  number  of  repeti- 
tions. 

The  latter  method  is  recommended  by  Moore  and  Seely,  and 
the  method  of  procedure  is  to  multiply  the  number  of  repetitions 
a  machine  is  to  withstand  by  the  factor  of  safety,  and  then  deter- 
mine the  endurance  stress  for  this  new  number  of  repetitions. 


TEMPERATURE  STRESSES 

22.  Deformation  Due  to  Temperature  Change. — It  is  important 
that  certain  machine  members  be  so  designed  that  expansion  as 


22 


TEMPERATURE  STRESSES 


[CHAP.  1 


well  as  contraction  due  to  a  change  in  temperature  may  take  place 
without  unduly  stressing  the  material.  Now  before  we  can  de- 
termine the  magnitude  of  such  stresses,  we  must  arrive  at  the 
deformation  caused  by  the  rise  or  drop  in  temperature.  The 
amount  that  a  member  will  change  in  length  depends  upon  the 
material  and  the  change  in  temperature,  and  may  be  expressed  by 
the  following  formula: 

A  =  atL,  (36) 

in  which  L  represents  the  original  length,  t  the  change  in  tempera- 
ture  in   degrees   Fahren- 
TABLE  4.— VALUES  OF  COEFFICIENT  OF      heitj   and   a  the  coefficient 

of  linear  expansion.     For 
values  of  a  consult  Table  4. 


LINEAR  EXPANSION 


Material 

Range  of 
temperature 

Coefficient  a. 

Cast  iron  

32  to  212 

0.00000618 

Wrought  iron  < 

32  to  212 
32  to  572 

0.00000656 
0.00000895 

Steel  casting.  .  .  . 

32  to  212 

0.00000600 

Soft  steel  

32  to  212 

0.00000630 

Nickel  steel  

32  to  212 

0.00000730 

Brass  casting.  .  . 

32  to  212 

0.0000104 

Bronze 

32  to  212 

0  0000100 

Copper.  .        .  \ 

32  to  212 

0.00000955 

32  to  572 

0.00001092 

23.  Stress  Due  to  Tem- 
perature  Change . —  Due 

to  the  deformation  A  dis- 
cussed in  the  preceding 
article,  the  machine  mem- 
ber subjected  to  a  change 
in  temperature  will  be 
stressed,  if  its  ends  are 
constrained  so  that  no 
expansion  or  contraction 
may  occur.  Knowing  the 

magnitude   of   A,  the  unit   strain  is  ^  from    which    we    may 

Li 

readily  determine  the  intensity  of  stress  due  to  a  change  t  in 
temperature,  by  applying  the  definition  of  the  modulus  of  elas- 
ticity given  in  Art.  5;  hence 

S  =  atE  (37) 


WORKING  STRESSES 

24.  Factor  of  Safety. — In  general,  the  maximum  stress  in- 
duced in  a  machine  part  must  be  kept  well  within  the  elastic 
limit  so  that  the  action  of  the  external  forces  is  almost  perfectly 
elastic.  The  stress  thus  used  in  arriving  at  the  size  of  the  part 
is  called  the  working  stress,  and  its  magnitude  depends  upon  the 
following  conditions: 

(a)  Is  the  application  of  load  steady  or  variable? 


ART.  24] 


WORKING  STRESSES 


23 


(6)  Is  the  part  subjected  to  unavoidable  shocks  or  jars? 

(c)  Kind  of  material,  whether  cast  iron,  steel,  etc. 

(d)  Is  the  material  used  in  the  construction  reliable? 

(e)  Is  human  life  or  property  endangered,  in  case  any  part  of 
a  machine  fails? 

(/)  In  case  of  failure  of  any  part,  will  any  of  the  remaining 
parts  of  the  machine  be  overloaded? 

(g)  Is  the  material  of  the  machine  part  subjected  to  unneces- 
sary and  speedy  deterioration? 

(h)  Cost  of  manufacturing. 

(i)  The  demand  upon  the  machine  at  some  future  time. 

As  usually  determined,  the  working  stress  for  a  given  case  is 
obtained  by  dividing  the  ultimate  strength  by  the  so-called 
factor  of  safety,  which  factor  should  really  represent  a  product  of 
several  factors  depending  upon  the  various  conditions  enumerated 
above.  In  general,  larger  factors  of  safety  are  used  when  a  piece 
is  made  of  cast  metal,  than  when  a  hammered  or  rolled  material 
is  used.  The  selection  of  a  larger  factor  of  safety  for  cast  metals 
is  due  to  the  fact  that  cast  parts  may  contain  hidden  blow  holes 
and  spongy  places.  In  many  cases  the  material  may  be  stressed 
an  unknown  amount  due  to  unequal  cooling  caused  by  the  im- 
proper distribution  of  the  material,  no  matter  how  careful  the 
moulder  may  be  in  cooling 
the  casting  after  it  is  poured. 

Again,  live  loads  require 
much  larger  factors  of  safety 
than  dead  loads,  and  loads 
that  produce  repetitive 
stresses  that  change  con- 


pression,  for  example,  also 
require  large  factors  of 
safety,  the  magnitudes  of 
which  are  difficult  to  deter- 


TABLE  5. — SUGGESTED  FACTORS  OF 
SAFETY 


Ki 

nd  of  stres 

s 

Material 

Steady 

Varying 

Shock 

Hard  steel  

5 

6 

15 

Structural  steel. 

4 

6 

10 

Wrought  iron.  . 
Cast  iron  

4 
6 

6 
10 

10 
20 

Timber  

6 

10 

15 

and 


mine.     For  the  latter  case,  the  equations  of  Arts.  19,  20 
21  may  serve  as  guide. 

In  Table  5  are  given  suggested  factors  of  safety  based  on  the 
ultimate  strength  of  the  material.  It  must  be  remembered  that 
the  skill  and  judgment  of  the  designer  should  play  an  important 
part  in  arriving  at  the  proper  working  stresses  for  any  given  set 
of  conditions. 


24 


TABLE  OF  PHYSICAL  PROPERTIES 


[CHAP.  I 


§O  O  O  O  O 
O  O  O  O  O 
O  O  O  O  O 


O  COCO-^TFCO  CO  1C 


si 


§1   I   i 


§ 


CD    00  O5  tVOOOiOC^C^^COS^^O  "^  OS    O 

<N     <N  (N  <NCS)(NCOCCr-lrH  i-Hr-l  ^H  t-H 


1C    CO    t^    (N 


O    >0    O    O 
Tf    CD    00    05 


§§ 


=3     8      I 

S     ^       £ 


2  3  § 

1J-! 


O  tf  O 


O 


S    2 

«    PQ 


ART.  24]  REFERENCES  25 

For  ultimate  strengths  and  various  other  physical  properties 
of  the  more  common  metals  used  in  the  construction  of  machinery, 
consult  Table  6. 

References 

Mechanics  of  Materials,  by  MERRIMAN. 
The  Strength  of  Materials,  by  E.  S.  ANDREWS. 
Mechanical  Engineers'  Handbook,  by  L.  S.  MARKS. 
Elasticitat  und  Festigkeit,  by  C,  BACH. 


CHAPTER  II 

MATERIALS  USED  IN  THE  CONSTRUCTION  OF 
MACHINE  PARTS 

The  principal  materials  used  in  the  construction  of  machine 
parts  are  cast  iron,  malleable  iron,  steel  casting,  steel,  wrought 
iron,  copper,  brass,  bronze,  aluminum,  babbitt  metal,  wood  and 
leather. 

CAST  IRON 

25.  Cast  Iron. — Cast  iron  is  more  commonly  used  than  any 
other  material  in  making  machine  parts.  This  is  because  of  its 
high  compressive  strength  and  because  it  can  be  given  easily 
any  desired  form.  A  wood  or  metal  pattern  of  the  piece  desired 
is  made,  and  from  this  a  mould  is  made  in  the  sand.  The  pattern 
is  next  removed  from  the  mould  and  the  liquid  metal  is  poured 
in,  which  on  cooling  assumes  the  form  of  the  pattern. 

Crude  cast  iron  is  obtained  directly  from  the  melting  of  the 
iron  ore  in  the  blast  furnace.  This  product  is  then  known  as 
pig  iron,  and  is  rarely  ever  used  except  to  be  remelted  into  cast 
iron,  or  to  be  converted  into  wrought  iron  or  steel.  Cast  iron 
fuses  easily,  but  it  cannot  be  tempered  nor  welded  under  ordinary 
conditions.  The  composition  of  cast  iron  varies  considerably, 
but  in  general  is  about  as  follows: 

Per  cent. 

Metallic  iron 90. 0    to  95 . 0 

Carbon 1.5    to    4.5 

Silicon 0.5    to    4.0 

Sulphur less  than  0 . 15 

Phosphorus 0. 06  to    1 . 50 

Manganese trace  to  5.0 

(a)  Carbon. — Carbon  may  either  be  united  chemically  with 
the  iron,  in  which  case  the  product  is  known  as  white  iron,  or  it 
may  exist  in  the  free  state,  when  the  product  is  known  as  gray 
iron.  The  white  iron  is  very  brittle  and  hard,  and  is  therefore 
but  little  used  in  machine  parts.  In  the  free  state  the  carbon 
exists  as  graphite. 

26 


ART.  26]  CAST  IRON  27 

(b)  Silicon. — Silicon  is  an  important  constituent  of  cast  iron 
because  of  the  influence  it  exerts  on  the  condition  of  the  carbon 
present  in  the  iron.     The  presence  of  from  0.25  to  1.75  per  cent, 
of  silicon  tends  to  make  the  iron  soft  and  strong;  but  beyond  2.0 
per  cent,  silicon,  the  iron  becomes  weak  and  hard.     An  increase 
of  silicon  causes  less  shrinkage  in  the  castings,  but  a  further  in- 
crease (above  5  per  cent.)  may  cause  an  increase  in  the  shrinkage. 
With  about  1.0  per  cent,  silicon,  the  tendency  to  produce  blow 
holes  in  the  castings  is  reduced  to  a  minimum. 

(c)  Sulphur. — Sulphur  in  cast  iron  causes  the  carbon  to  unite 
chemically  with  the  iron,  thus  producing  hard  white  iron,  which 
is  brittle.     For  good  castings,  the  sulphur  content  should  not 
exceed  0.15  per  cent. 

(d)  Phosphorus. — Phosphorus  in   cast  iron  tends  to  produce 
weak  and  brittle  castings.     It  also  causes  the  metal  to  be  very 
fluid  when  melted,  thus  producing  an  excellent  impression  of  the 
mould.     For  this  reason  phosphorus  is  a  desirable  constituent 
in  cast  iron  for  the  production  of  fine,  thin  castings  where  no 
great  strength  is  required.     To  produce  such  castings,  from  2  to  5 
per  cent,  of  phosphorus  may  be  used.     For  strong  castings  of 
good  quality,  the  amount  of  phosphorus  rarely  exceeds  0.55  per 
cent.,  but  when  fluidity  and  softness  are  more  important  than 
strength,  from  1  to  1.5  per  cent,  may  be  used. 

(e)  Manganese. — Manganese  when  present  in  cast  iron  up  to 
about  1.5  per  cent,  tends  to  make  the  castings  harder  to  machine; 
but  renders  them  more  suitable  for  smooth  or  polished  surfaces. 
It  also  causes  a  fine  granular  structure  in  the  castings  and  pre- 
vents  the   absorption   of  the   sulphur   during  melting.     Man- 
ganese may  also  be  added  to  cast  iron  to  soften  the  metal.     This 
softening  is  due  to  the  fact  that  the  manganese  counteracts  the 
effects  of  the  sulphur  and  silicon  by  eliminating  the  former  and 
counteracting  the  latter.     However,  when  the  iron  is  remelted, 
its  hardness  returns  since  the  manganese  is  oxidized  and  more 
sulphur  is  absorbed.     The  transverse  strength  of  cast  iron  is 
increased  about  30  per  cent.,  and  the  shrinkage  and  depth  of 
chill  decreased  25  per  cent.,  while  the  combined  carbon  is  dimin- 
ished one-half  by  adding  to  the  molten  metal,  powdered  ferro- 
manganese  in  the  proportion  of  1  pound  of  the  latter  to  about 
600  pounds  of  the  former. 

26.  Vanadium  Cast  Iron. — The  relatively  coarse  texture  of 
cast  iron  may  be  much  improved  by  the  addition  of  0.10  to  0.20 


28 


PIG  IRON 


[CHAP.  II 


per  cent,  of  vanadium,  and  at  the  same  time  the  ultimate  strength 
is  increased  from  10  to  25  per  cent.  Cast  iron  containing  a  small 
percentage  of  vanadium  is  tougher  than  ordinary  gray  iron,  thus 
making  it  an  excellent  material  for  use  in  steam-  and  gas-engine 
cylinders,  piston  rings,  liners,  gears  and  other  similar  uses.  Some 
of  the  larger  railway  systems  have  now  adopted  this  material  for 
their  cylinder  construction.  In  machining  vanadium  cast  iron, 
it  is  possible  to  give  it  a  much  higher  finish  than  is  possible  with 
gray  iron. 

27.  Pig  Iron. — Pig  iron  is  the  basis  for  the  manufacture  of  all 
iron  products.  It  is  not  only  used  practically  unchanged  to  pro- 
duce castings  of  a  great  variety  of  form  and  quality,  but  it  is  also 
used  in  the  manufacture  of  wrought  iron  and  steel.  For  each 
special  purpose,  the  iron  must  have  a  composition  within  certain 
limits.  It  follows,  therefore,  that  pig  iron  offers  a  considerable 
variety  of  composition.  The  practice  of  purchasing  pig  iron 
by  analysis  is  generally  followed  at  the  present  time.  In  Table 
7  are  give  the  specifications  for  the  various  grades  of  pig  iron 
used  by  one  large  manufacturer. 

TABLE  7. — SPECIFICATIONS  OF  PIG  IRON 


Class 

Total  carbon 
not  under, 
per  cent. 

Silicon, 
per  cent. 

Sulphur 
not  over, 
per  cent. 

Phosphorus, 
per  cent. 

Manganese 
not  over, 
per  cent. 

1 

3.0 

1  .  5  to  2  .  0 

0.040 

0.20  to  0.75 

1.0 

2 

3.5 

2.0to2.5 

0.035 

0.20  to  0.75 

1.0 

3 

3.5 

2  .  5  to  3  .  0 

0.030 

0.20  to  0.75 

1.0 

4 

3.5 

2.0to2.5 

0.040 

1.00  to  1.50 

1.0 

5 

3.0 

4.0to5.0 

0.040 

0.20  to  0.80 

1.0 

In  general,  an  analysis  is  made  from  drillings  taken  from  a  pig 
selected  at  random  from  each  four  tons  of  every  carload  as  un- 
loaded. The  right  is  reserved  to  reject  a  portion  or  all  of  the 
material  which  does  not  conform  to  the  above  specifications  in 
every  particular. 

In  a  general  way,  the  specified  limits  for  the  composition  of  the 
chief  grades  of  pig  iron  are  given  in  Table  8. 

According  to  use,  pig  iron  may  be  divided  roughly  into  two 
classes.  The  first  class  includes  those  grades  used  in  the  produc- 
tion of  foundry  and  malleable  irons,  while  the  second  includes 
those  used  in  the  manufacture  of  wrought  iron  and  steel.  In  the 
process  of  remelting  or  manufacturing,  the  first  class  undergoes 


ART.  28] 


CHILLED  CASTING 


29 


little  if  any  chemical  change,  while  the  second  class  undergoes 
a  complete  chemical  change. 

TABLE   8. — GENERAL   SPECIFICATIONS   OF  PIG  IRON 


Grade  of  iron 

Silicon, 
per  cent. 

Sulphur, 
per  cent. 

Phosphorus, 
per  cent. 

Manganese, 
per  cent. 

No   1  foundry 

2.5to3.0 
2.0to2.5 
1.5  to2.0 
0.7  to  1.5 
Under  1.5 
1.0to2.0 
Under  2.0 
Under  1  .  0 
Under  1  .  0 

Under  0.035 
Under  0.045 
Under  0.055 
Under  0.050 
Under  0.100 
Under  0.050 
Under  0.030 
Under  0.050 
Under  0.050 

0.5  to  1.0 

Under  0.2 
Under  1  .  0 
Under  0.1 
Under  0.3 
Under  1  .  0 
2.0to3.0 

Under  1  .  0 
1  .  0  to  2  .  0 

No.  2  foundry  
No   3  foundry.. 

Malleable  

Gray  forge..              .        .... 

Bessemer  

Low  phosphorus  

Basic 

Basic  Bessemer  

28.  Malleable  Casting. — Malleable  castings  are  made  by  heat- 
ing clean  foundry  castings,  preferably  with  the  sulphur  content 
low,  in  an  annealing  furnace  in  contact  with  some  substance  that 
will  absorb  the  carbon  from  the  cast  iron.     Hematite  or  brown 
iron  ore  in  pulverized  form  is  used  extensively  for  that  purpose. 
The  intensity  of  heat  required  is  on  the  average  about  1,650°F. 
The  length  of  time  the  castings  remain  in  the  furnace  depends 
upon  the  degree  of  malleability  required  and  upon  the  size  of  the 
castings.     Usually  light  castings  require  a  minimum  of  60  hours, 
while  the  heavier  ones  may  require  72  hours  or  longer. 

The  tensile  strength  of  good  malleable  cast  iron  lies  somewhere 
between  that  of  gray  iron  and  steel,  while  its  compressive  strength 
is  somewhat  lower  than  that  of  the  former.  Good  malleable  cast- 
ings may  be  bent  and  twisted  without  showing  signs  of  fracture, 
and  for  that  reason  are  well  adapted  for  use  in  connection  with 
agricultural  machinery,  railroad  supplies,  and  automobile  parts. 

29.  Chilled  Casting. — Chilled  castings  are  those  which  have  a 
hard  and  durable  surface.    The  iron  used  is  generally  close-grained 
gray  iron  low  in  silicon.     A  chilled  casting  is  formed  by  making 
that  part  of  the  mould  in  contact  with  the  surface  of  the  casting 
to  be  chilled  of  such  construction  that  the  heat  will  be  with- 
drawn rapidly.     The  mould  for  causing  the  chill  usually  con- 
sists of  iron  bars  or  plates,  placed  so  that  their  surfaces  will  be  in 
contact  with  the  molten  iron.     These  plates  abstract  heat  rapidly 
from  the  iron,  with  the  result  that  the  part  of  the  casting  in  con- 


30  WROUGHT  IRON  [CHAP.  II 

tact  with  the  cold  surface  assumes  a  state  similar  to  white  iron, 
while  the  rest  of  the  casting  remains  in  the  form  of  gray  iron. 
The  withdrawal  of  heat  is  hastened  by  the  circulation  of  cold 
water  through  pipes,  circular  or  rectangular  in  cross-section,  placed 
near  the  surface  to  be  chilled.  Chilled  castings  offer  great  re- 
sistance to  crushing  forces.  The  outside  or  "skin"  of  the  ordi- 
nary casting  is  in  fact  a  chilled  surface,  but  by  the  arrangement 
mentioned  above,  the  depth  of  the  "skin"  is  greatly  increased 
with  a  corresponding  increase  in  strength  and  wearing  qualities. 
Car  wheels,  jaws  for  crushing  machinery,  and  rolls  for  rolling 
mills  are  familiar  examples  of  chilled  castings.  Car  wheels  re- 
quire great  strength  combined  with  a  hard  durable  tread.  The 
depth  of  the  chill  varies  from  %  to  1  inch. 

It  has  been  found  that  with  the  use  of  vanadium  in  chilled 
castings,  a  deeper,  stronger  and  tougher  chill  can  be  produced. 
This  chill,  however,  is  not  quite  as  hard  as  that  found  on  ordinary 
chilled  cast  iron,  and  hence  has  the  advantage  that  such  castings 
can  be  filed  and  machined  more  easily. 

30.  Semi-steel. — The  term  semi-steel  is  applied  to  a  metal  that 
is   intermediate   between   cast  iron   and   malleable  iron.     The 
meaning  of  the  term  as  used  at  the  present  time  is  vague  and  for 
that  reason  its  use  is  questioned.     The  so-called  semi-steel  is 
produced  in  the  cupola  by  mixing  from  20  to  40  per  cent,  of  low- 
carbon  steel  scrap  with  the  pig  iron  and  cast  scrap.     This  mix- 
ture, if  properly  handled  in  the  cupola  as  well  as  in  pouring  the 
mould,  produces  a  clean  close-grained  tough  casting  that  may  be 
machined  easily  and  that  has  an  ultimate  tensile  strength  vary- 
ing from  32,000  to  42,000  pounds  per  square  inch.     Its  trans- 
verse strength  is  also  considerably  higher  than  that  of  ordinary 
gray  iron.     However,  the  material  produced  by  such  a  mixture 
as  given  above  has  none  of  the  distinctive  properties  of  steel  and 
in  reality  it  is  nothing  more  than  a  high-grade  gray-iron  casting. 
Semi-steel  has  been  used  very  successfully  for  cylinders,  piston 
rings,  cylinder  liners,  gears,  plow  points,  and  frames  of  punches 
and  shears. 

WROUGHT  IRON 

31.  Wrought  Iron. — Wrought  iron  is  formed  from  pig  iron  by 
melting  the  latter  in  a  puddling  furnace.     During  the  process  of 
melting,  the  impurities  in  the  pig  iron  are  removed  by  oxidation, 
leaving  the  pure  iron  and  slag  both  in  a  pasty  condition.     In  this 


ART.  32]  STEEL  CASTING  31 

condition  the  mixture  of  iron  and  slag  is  formed  into  muck  balls 
weighing  about  150  pounds,  and  is  removed  from  the  furnace 
These  balls  are  put  into  a  squeezer  and  compressed,  thereby  re- 
moving a  large  amount  of  the  slag,  after  which  it  is  rolled  into 
bars.  The  bars,  known  as  "muck  bars,"  are  cut  into  strips  and 
arranged  in  piles,  the  strips  in  the  consecutive  layers  being  at 
right  angles  to  each  other.  These  piles  are  then  placed  into  a 
furnace  and  raised  to  a  welding  heat  and  are  then  rolled  into  mer- 
chant bars.  If  the  quality  of  the  iron  is  to  be  improved  and  the 
last-mentioned  process  is  repeated,  we  obtain  what  is  known  as 
"best  iron,"  "double  best"  and  "treble  best,"  depending  upon  the 
number  of  repetitions.  The  merchant  bar  finally  produced  is 
the  ordinary  wrought  iron  of  commerce.  At  the  present  time 
wrought  iron  is  not  used  as  extensively  as  in  the  past,  steel  to  a 
great  extent  having  taken  its  place;  however,  it  still  is  used  in  the 
manufacture  of  pipes,  boiler  tubes,  forgings,  parts  of  electrical 
machinery,  small  structural  shapes,  and  crucible  steel. 

STEEL  CASTING 

32.  Manufacturing  Processes. — Castings  similar  to  iron  cast- 
ings may  be  formed  in  almost  any  desired  shape  from  molten 
steel.  They  are  produced  by  four  distinct  methods  as  follows: 

(a)  Crucible  process. — When  it  is  desired  to  produce  very  fine 
and  high-grade  castings,  not  very  large,  the  crucible  process  is  used. 

(6)  Bessemer  process. — This  method  is  used  chiefly  for  pro- 
ducing small  castings. 

(c)  Open-hearth  process. — The  open-hearth  process  is  used  ex- 
tensively for  the  production  of  steel  castings  either  small  or  ex- 
tremely large  in  size.     The  castings  produced  by  this  method  are 
considered  superior  to  those  produced  by  the  Bessemer  process. 

(d)  Electric-furnace  method. — The  electric  furnace  which  is  now 
being  introduced  into  this  country  is  capable  of  producing  the 
very  best  grades  of  steel  castings. 

In  texture,  the  castings  produced  by  the  common  processes  in 
use  today  are  coarse  and  crystalline,  since  the  steel  has  been  per- 
mitted to  cool  without  drawing  or  rolling.  In  order  to  improve 
the  grain  structure,  and  at  the  same  time  remove  some  of  the 
internal  stresses,  all  steel  castings  must  be  annealed  before  ma- 
chining them.  Formerly  trouble  was  experienced  in  obtaining 
good  sound  steel  castings ;  but  by  great  care  and  improved  meth- 


32  MANGANESE-STEEL  CASTINGS  [CHAP.  II 

ods  in  the  production  of  moulds,  first-class  castings  may  now 
be  obtained.  In  general,  steel  castings  are  used  for  those  machine 
parts  requiring  greater  strength  than  is  obtained  by  using  gray- 
iron  castings. 

33.  Manganese-steel  Castings. — Manganese-steel  castings  are 
produced  by  adding  ferro-manganese  to  open-hearth  steel,  and  the 
average  chemical  composition  of  such  castings  is  about  as  follows: 
Manganese,  12.5  per  cent.;  carbon,  1.25  per  cent.;  silicon,  0.3  per 
cent.;  phosphorus,  0.08  per  cent.;  sulphur,  0.02  per  cent.;  iron, 
85.85  per  cent.  The  average  physical  properties  of  this  kind  of 
steel  casting  are  about  as  follows: 

Tensile  strength 110,000  pounds  per  square  inch. 

Elastic  limit 54,000  pounds  per  square  inch. 

Elongation  in  8  inches. ...  45  per  cent. 

Reduction  of  area 50  per  cent. 

Manganese  steel  is  in  general  free  from  blow  holes,  but  is  diffi- 
cult to  cast  on  account  of  its  high  shrinkage,  which  is  about  two 
and  one-half  times  as  great  as  that  of  cast  iron.  As  originally 
cast  it  is  extremely  hard  and  brittle  and  it  is  possible  to  pulverize 
it  under  the  blows  of  a  hammer.  The  fact  that  this  metal  is 
brittle  when  it  comes  from  the  mould  makes  it  possible  to  break 
off  the  risers  and  gates  remaining  on  the  casting,  which  could 
not  be  done  were  the  original  casting  as  tough  as  the  finished 
product.  As  mentioned,  manganese-steel  casting  possesses  great 
hardness  which  is  not  diminished  by  annealing,  and  in  addition 
it  has  a  high  tensile  strength  combined  with  great  toughness  and 
ductility.  These  qualities  would  make  this  steel  the  ideal  metal 
for  machine  construction,  were  it  not  for  the  fact  that  its  great 
hardness  prevents  it  from  being  machined  in  any  way  but  by 
abrasive  processes,  which  at  best  are  expensive.  Again,  the 
very  property  of  hardness,  combined  with  great  toughness,  also 
limits  its  use  to  the  rougher  class  of  castings,  or  such  that  require 
a  minimum  amount  of  finish. 

The  toughness  of  the  finished  casting  is  produced  by  the  an- 
nealing process.  In  this  process  the  brittle  castings  are  placed 
in  annealing  furnaces,  in  which  they  are  heated  gradually  and 
carefully.  After  remaining  in  these  furnaces  from  three  to 
twenty-four  hours,  depending  upon  the  type  of  casting  treated, 
the  castings  are  removed  from  the  furnace  and  quenched  in  cold 
water.  It  is  evident  that  great  care  must  be  exercised  by  the 


ART.  34]  MANGANESE-STEEL  CASTINGS  33 

designer  to  distribute  the  metal  properly  in  large  and  complicated 
castings  in  order  that  all  the  parts  may  cool  at  approximately 
the  same  rate.  It  has  been  found  by  experience  that  the  heat 
treatment  just  described  cannot  be  made  to  extend  through  a 
section  thicker  than  5  or  5J/2  inches.  In  general,  thicknesses 
exceeding  3^  inches  are  not  found  in  well-designed  casting. 

34.  Applications  of  Manganese-steel  Castings. — Due  to  the 
fact  that  manganese-steel  casting  is  the  most  durable  metal  known 
as  regards  ability  to  resist  wear,  it  is  well  adapted  to  the  following 
classes  of  service : 

(a)  For  all  wearing  parts  of  crushing  and  pulverizing  machinery, 
such  as  rolls,  jaws  and  toggle  plates,  heads,  mantels,  and  concaves. 

(6)  In  all  classes  of  excavating  machinery;  for  example,  the 
dipper  and  teeth  of  dipper  dredges,  the  buckets  of  placer  dredges, 
the  cutter  head  and  knives  of  ditching  machines. 

(c)  The  impellers  and  casings  of  centrifugal  pumps  are  fre- 
quently made  of  manganese-steel  casting.     In  this  connection  it 
is  of  interest  to  note  that  soft-steel  inserts  are  cast  into  the  casing 
at  proper  places  to  permit  the  drilling  and  tapping  of  holes  for 
the  various  attachments. 

(d)  In  connection  with  hoisting  machinery  such  parts  as  sheaves, 
drums,  rollers,  and  crane  wheels  made  of  manganese-steel  cast- 
ing are  not  uncommon.     It  is  claimed  that  the  life  of  a  rope 
sheave  or  roller  made  of  this  material  is  about  thirty  times  that 
of  one  made  of  cast  iron. 

(e)  In  mining  work,  the  wheels  of  coal  cars  and  skips,  also  the 
head  sheaves,  are  made  of  manganese  steel.     In  the  latter  appli- 
cation, the  rim  only  is  made  of  manganese  steel  and  is  then 
bolted  to  the  wrought-iron  spokes,  which  in  turn  are  bolted  to 
the  cast-iron  hub. 

(/)  In  conveying  machinery  where  the  parts  are  subjected  to 
severe  usage,  as  for  example  a  conveyor  chain  in  a  cement  mill, 
both  the  chains  and  the  sprockets  are  made  of  manganese-steel 
casting. 

(g)  In  railway  track  work,  manganese-steel  casting  has  given 
excellent  service  for  crossings,  frogs,  switches,  and  guard  rails. 

(h)  Another  very  important  use  of  manganese-steel  casting 
is  in  the  construction  of  safes  and  vaults;  for  this  purpose  it  is 
particularly  well  adapted  since  it  cannot  be  drilled  nor  can  its 
temper  be  drawn  by  heating. 


34  BESSEMER  PROCESS  [CHAP.  II 

STEEL 

Steel  is  a  compound  in  which  iron  and  carbon  are  the  principal 
parts.  It  is  made  from  pig  iron  by  burning  out  the  carbon, 
silicon,  manganese  and  other  impurities,  and  recarbonizing  to 
any  degree  desired.  The  principal  processes  or  methods  of 
manufacturing  steel  are  the  following:  (a)  the  Bessemer;  (6) 
the  open-hearth;  (c)  the  cementation. 

35.  Bessemer  Process. — In  the  Bessemer  process,  several  tons, 
usually  about  ten,  of  molten  pig  iron  are  poured  into  a  con- 
verter, and  through  this  mass  of  iron  a  large  quantity  of  cold  air 
is  passed.     In  about  four  minutes  after  the  blast  is  turned  on, 
all  the  silicon  and  manganese  of  the  pig  iron  has  combined  with 
the  oxygen  of  the  air.     The  carbon  in  the  pig  iron  now  begins 
to  unite  with  the  oxygen,  forming  carbon  monoxide,  which  burns 
through  the  mouth  of  the  converter  in  a  long  brilliant  flame. 
The  burning  of  the  carbon  monoxide  continues  for  about  six 
minutes,  when  the  flame  shortens,  thus  indicating  that  nearly 
all  of  the  carbon  has  been  burned  out  of  the  iron  and  that  the 
air  supply  should  be  shut  off.     The  burning  out  of  these  impuri- 
ties has  raised  the  temperature  of  the  iron  to  a  white  heat,  and 
at  the  same  time  produced  a  relatively  pure  mass  of  iron.     To 
this  mass  is  added  a  certain  amount  of  carbon  in  the  form  of  a 
very  pure  iron  high  in  carbon  and  manganese.     The  metal  is 
then  poured  into  moulds  forming  ingots,  which  while  hot  are 
rolled  into  the  desired  shapes. 

The  characteristics  of  the  Bessemer  process  are:  (a)  great 
rapidity  of  reduction,  about  ten  minutes  per  heat;  (6)  no  extra 
fuel  is  required;  (c)  the  metal  is  not  melted  in  the  furnace  where 
the  reduction  takes  place. 

Bessemer  steel  was  formerly  used  almost  entirely  in  the  manu- 
facture of  wire,  skelps  for  tubing,  wire  nails,  shafting,  machinery 
steel,  tank  plates,  rails,  and  structural  shapes.  Open-hearth 
steel,  however,  has  very  largely  superseded  the  Bessemer  product 
in  the  manufacture  of  these  articles. 

36.  Open-hearth  Process. — In  the  manufacture  of  open-hearth 
steel,  the  molten  pig  iron,  direct  from  the  reducing  furnace,  is 
poured  into  a  long  hearth,  the  top  of  which  has  a  firebrick  lining. 
The  impurities  in  the  iron  are  burned  out  by  the  heat  obtained 
from  burning  gas  and  air,  and  reflected  from  this  refractory  lining. 


ART.  37]  CRUCIBLE  STEEL  35 

The  slag  is  first  burned,  and  the  slag  in  turn  oxidizes  the  im- 
purities. The  time  required  for  purifying  is  from  6  to  10  hours, 
after  which  the  metal  is  recarbonized,  cast  into  ingots-  and  rolled 
as  in  the  Bessemer  process. 

The  characteristics  of  the  open-hearth  process  are :  (a)  relatively 
long  time  to  oxidize  the  impurities;  (6)  large  quantities,  35  to 
70  tons,  may  be  purified  and  recarbonized  in  one  charge;  (c) 
extra  fuel  is  required;  (d)  a  part  of  the  charge,  steel  scrap  and 
iron  ore  added  at  the  beginning  of  the  process,  are  melted  in  the 
furnace. 

Open-hearth  steel  is  used  in  the  manufacture  of  cutlery,  boiler 
plate,  and  armor  plate  in  addition  to  the  articles  mentioned  in 
Art.  35. 

37.  Cementation  Process. — In  this  process  of  manufacturing 
steel,  bars  of  wrought  iron  imbedded  in  charcoal  are  heated  for 
several  days.     The  wrought  iron  absorbs  carbon  from  the  char- 
coal and  is  thus  transformed  into  steel.     When  the  bars  of  iron 
are  removed  they  are  found  to  be  covered  with  scales  or  blisters. 
The  name  given  to  this  product  is  blister  steel.     By  removing  the 
scales  and  blisters  and  subjecting  the  bars  to  a  cherry-red  heat 
for  a  few  days,  a  more  uniform  distribution  of  the  carbon  is 
obtained. 

Blister  steel  when  heated  and  rolled  directly  into  the  finished 
bars,  is  known  as  German  steel.  Bars  of  blister  steel  may  be  cut 
up  and  forged  together  under  the  hammer,  forming  a  product 
called  shear  steel.  By  repeating  the  process  with  the  shear  steel, 
we  obtain  double-shear  steel. 

38.  Crucible  Steel. — Crucible  steel,  also  called  cast  steel,  is 
very  uniform  and  homogeneous  in  structure.     It  is  made  by 
melting  blister  steel  in  a  crucible,  casting  it  into  ingots  and  rolling 
into  bars.     By  this  method  is  produced  the  finest  crucible  steel. 
Another  method   of   producing   crucible-cast   steel   is   to   melt 
Swedish  iron  (wrought  iron  obtained  from  the  reduction  of  a 
very  pure  iron  in  the  blast  furnace  in  which  charcoal  instead  of 
coke  for  producing  the  puddling  flame  is  used)  in  contact  with 
charcoal  in  a  sealed  vessel,  the  contents  of  which  are  poured 
into  a  large  ladle  containing  a  similar  product  from  other  sealed 
vessels.     This  mixing   insures  greater  uniformity  of  material. 
The  metal  in  this  large  ladle  is  cast  into  ingots,  which  are  sub- 


36  NICKEL  STEEL  [CHAP.  II 

sequently  forged  or  rolled  into  bars.     By  far  the  greater  part 
of  crucible  steel  is  produced  by  this  method. 

39.  Cold-rolled  Steel. — The  so-called  cold-rolled  steel  is  rolled 
hot  to  approximately  the  required  dimensions.     The  surface  is 
then  carefully  cleaned,  usually  by  chemical  means,  and  rolled 
cold  to  a  very  accurately  gauged  thickness  between  smooth 
rollers.     The  rolling  of  metal  when   cold  has  two  important 
advantages  as  follows:  when  steel  is  rolled  hot  the  surface  of 
the  steel  oxidizes  and  forms  a  scale,  while  with  cold  rolling  no 
such  action  takes  place,  thus  making  it  possible  to  produce  a 
bright  finish.     Furthermore,  since  no  scale  is  formed  the  bar  or 
plate  to  be  rolled  can  be  made  very  accurate.     The  cold-rolling 
process  has  the  effect  of  increasing  the  elastic  limit  and  ultimate 
strength,  but  decreases  the  ductility.     It  also  produces  a  very 
smooth  and  hard  surface.     Its  principal  use  is  for  shafting  and 
rectangular,  square  and  hexagonal  bars,  as  well  as  strip  steel 
which  of  late  is  in  demand  for  use  in  the  manufacture  of  pressed- 
steel  products.     For  the  latter  class  of  work  the  absence  of  scale, 
already  referred  to,  has  a  marked  effect  on  the  life  of  the  dies, 
as  experience  in  press  working  of  hot-rolled  metal  shows  that  the 
scale  on  the  latter  is  exceedingly  hard  on  the  dies. 

ALLOY  STEELS 

The  term  alloy  steels  is  applied  to  all  steels  that  are  composed 
of  iron  and  carbon,  and  one  or  more  special  elements  such  as 
nickel,  tungsten,  manganese,  silicon,  chromium,  and  vanadium. 
In  general,  alloy  steels  must  always  be  heat  treated,  and  should 
never  be  used  in  the  natural  or  annealed  condition,  since  in  the 
latter  condition  the  physical  properties  of  the  material  are  but 
little  better  than  those  of  the  ordinary  carbon  steels.  The  heat 
treatment  given  to  alloy  steels  causes  a  marked  improvement  in 
the  physical  properties.  A  few  of  the  principal  alloy  steels  are 
discussed  in  the  following  paragraphs. 

40.  Nickel  Steel. — Nickel  added  to  a  carbon  steel  increases  its 
ultimate  strength  and  elastic  limit  as  well  as  its  hardness  and 
toughness.     It  tends  to  produce  a  steel  that  is  more  homogeneous 
and  of  finer  structure  than  the  ordinary  carbon  steel,  and  if  the 
percentage  of  nickel  is  considerable  the  material  produced  resists 
corrosion  to  a  remarkable  degree.    The  percentage  of  nickel 


ART.  41]  VANADIUM  STEEL  37 

varies  from  1.5  to  4.5,  while  the  carbon  varies  from  0.15  to  0.50 
per  cent.,  both  of  these  percentages  depending  upon  the  grade 
of  steel  desired.  Nickel  steel  has  a  high  ratio  of  elastic  limit  to 
ultimate  strength  and  in  addition  offers  great  resistance  to  crack- 
ing. The  latter  property  makes  this  type  of  steel  desirable  for 
use  as  armor  plate.  Nickel  steel  is  also  used  for  structural 
shapes  and  for  rails;  the  latter  show  better  wearing  qualities 
than  those  made  from  Bessemer  or  open-hearth  steel.  On 
account  of  its  ability  to  withstand  heavy  shocks  and  torsional 
stresses,  nickel  steel  is  well  adapted  for  crankshafts,  high-grade 
shafting,  connecting  rods,  automobile  parts,  car  axles  and 
ordnance. 

41.  Chrome  Steel. — Chrome  steel  is  produced  by  adding  to 
high-carbon  steel  (0.8  to  2.0  per  cent.)  from  1  to  2  per  cent,  of 
chromium.     The  steel  thus  produced  is  very  fine-grained  and 
homogeneous,  is  extremely  hard,  and  has  a  high  ratio  of  elastic 
limit  to  ultimate  strength.     Due  to  its  extreme  hardness,  chrome 
steel  may  be  used  for  ball  and  roller  bearings,  armor-piercing 
shells,  armor  plate,  burglar-proof  safes,  and  vaults.     The  element 
chromium  is  also  used  in  the  manufacture  of  the  best  high-speed 
tool  steels. 

42.  Vanadium  Steel. — Vanadium  steel  is  produced  by  adding 
to  carbon  steel,  a  small  amount  of  vanadium,  generally  between 
0.15  and  0.25  per  cent.      This  alloy  steel  is  used  as  a  forging  or 
machinery  steel,  and  should  be  heated  slowly  when  preparing 
it  for  a  forging  operation.     The  effect  of  the  vanadium  is  to 
increase  the  elastic  limit  as  well  as  the  capacity  for  resisting  shock. 
Vanadium  is  used  more  in  conjunction  with  chromium  or  nickel 
steel  than  with  ordinary  carbon  steel.     Carbon  vanadium  steel 
containing  from  0.60  to  1.25  per  cent,  carbon  and  over  0.2  per 
cent,  vanadium  may  be  tempered,  and  due  to  its  toughness,  is 
well  adapted  for  punches,  dies,  rock  drills,  ball  and  roller  bearings, 
and  other  similar  uses. 

43.  Nickel-chromium  Steel. — Nickel-chromium  steel  is  used 
chiefly  in  automobile  construction,  where  a  high  degree  of  strength 
and  hardness  is  demanded.     At  the  present  time  this  type  of 
steel  is  also  being  used  for  important  gears  on  machine  tools. 
In  the  automobile  industry,  three  types  of  nickel-chromium  steels 
are  commonly  used.     These  are  known  as  low  nickel-,  medium 
nickel-,  and  high  nickel-chromium  steels. 


38  CHROMIUM-VANADIUM  STEEL  [CHAP.  II 

In  general,  nickel-chromium  steels  having  a  carbon  content 
up  to  0.2  per  cent,  are  intended  for  case  hardening;  those  having 
0.25  to  0.4  per  cent,  carbon  are  used  for  the  structural  parts  of 
automobiles,  while  the  higher-carbon  steels  may  be  used  for  gears 
or  other  important  parts. 

44.  Chromium-vanadium   Steel. — Chromium- vanadium   steel 
is  tough  and  capable  of  resisting  severe  shocks,  and  has  an  exceed- 
ingly high  elastic  limit  in  proportion  to  its  ultimate  strength. 
This  type-  of  steel  is  used  for  springs,  gears,  driving  shafts,  steering 
knuckles,  and  axles  in  the  automobile  industry.     It  is  also  used 
for  spindles  and  arbors  for  machine  tools,  locomotive  driving 
axles,  piston  rods,  side  and  connecting  rods,  and  locomotive  and 
car- wheel  tires. 

For  high-duty  shafts  requiring  a  high  degree  of  strength  and  a 
moderate  degree  of  toughness,  the  grade  of  chromium-vanadium 
steel  containing  about  0.4  per  cent,  carbon  should  be  selected. 
For  springs  and  gears  the  carbon  content  should  be  from  0.45  to 
0.50  per  cent.  Chromium -vanadium  steels  having  a  high  carbon 
content  of  0.75  to  1.0  per  cent,  may  be  tempered  and  used  for 
tools.  In  addition  to  being  hard  it  is  tough,  and  for  that  reason 
has  been  used  successfully  for  dies,  punches,  ball-bearing  races, 
rock  drills,  and  saws. 

45.  Silicon-manganese  Steel. — A  combination  of  silicon  and 
manganese  in  moderate  amounts  added  to  steel  increases  its 
capacity  for  resisting  shock,  thus  making  it  particularly  suitable 
for  all  kinds  of  springs  and  to  some  extent  for  gears.     For  each 
class  of  service  mentioned  the  steel  must  be  given  a  proper  heat 
treatment. 

46.  Tungsten  Steel. — Tungsten  steel  is  an  alloy  of  iron,  carbon, 
tungsten  and  manganese,  and  sometimes  chromium.     The  ele- 
ment which  gives  this  steel  its  peculiar  property,  self  or  air  harden- 
ing, is  not  tungsten  but  manganese  combined  with  carbon.     The 
tungsten,  however,  is  an  important  element,  since  it  enables  the 
alloy  to  contain  a  larger  percentage  of  carbon.     On  account  of  its 
hardness,  this  steel  can  not  be  easily  machined,  but  must  be 
forged  to  the  desired  shape.     Its  chief  use  is  for  high-speed  cutting 
tools. 

ALLOYS 

Alloys  may  be  made  of  two  or  more  metals  that  have  an  affinity 
for  each  other.  The  compound  or  alloy  thus  produced  has 


ART.  47]  BRASS  39 

properties  and  characteristics  which  none  of  the  metals  possess. 
The  principal  alloys  used  in  machine  construction  may  be  ob- 
tained by  combining  two  or  more  of  the  following  metals :  copper, 
zinc,  tin,  lead,  antimony,  bismuth,  and  aluminum. 

47.  Brass. — Brass  is  an  alloy  of   copper  and  zinc;  however, 
many  of  the  commercial  brasses  contain  small  percentages  of 
lead,  tin,  and  iron.     Brass  for  machine  parts  may  be  put  in  two 
general  classes,  namely,  cast  brass  and  wrought  brass. 

(a)  Cast  brass. — Cast  brass  is  intended  for  parts  not  requiring 
great  strength,  and  as  usually  made  has  a  zinc  content  of  about 
35  per  cent.,  and  the  remainder  copper  with  traces  of  iron,  lead 
and  tin.     In  order  to  make  cast  brass  free-cutting  for  machining 
purposes  1  to  2  per  cent,  of  lead  is  added.     A  typical  specification 
for  cast  brass  as  used  by  the  Bureau  of  Steam  Engineering  of  the 
United  States  Navy  Department  is  as  follows:  copper,  59  to  63 
per  cent.;  tin,  0.5  to  1.5  per  cent.;  iron,  not  exceed  0.06  per  cent.; 
lead,  not  exceed  0.60  per  cent.;  zinc,  remainder. 

(b)  Wrought   brass. — Wrought  brass  may  be  of  two  kinds  as 
follows:     1.  That  which  contains  approximately  56  to  62  per 
cent,  of  copper  and  the  remainder  zinc  may  be  rolled  or  forged 
while  hot.     Muntz  metal  containing  60  per  cent,  of  copper  and 
40  per  cent,  zinc  is  a  well-known  wrought  brass  which  at  one  time 
was  used  very  extensively  for  ship  sheathing.     The  so-called 
Tobin  bronze  is  another  type  of  wrought  brass  that  may  be 
worked  while  hot,  but  it  differs  from  Muntz  metal  in  that  it 
contains  very  small  percentages  of  iron,  tin,  and  lead,  in  addition 
to  the  copper  and  zinc.     Its  ultimate  tensile  strength  is  about 
equal  to  that  of  ordinary  steel,  while  its  compressive  strength  is 
about  three  times  its  tensile  strength.     Tobin  bronze  resists 
corrosion  and  for  that  reason  meets  with  favor  in  naval  work. 

2.  The  second  kind  of  wrought  brass  contains  approximately 
70  per  cent,  of  copper  and  30  per  cent,  of  zinc,  and  not  infre- 
quently a  small  percentage  of  lead  is  introduced  to  facilitate 
machining.  Brass  having  the  composition  just  stated  may  be 
drawn  or  rolled  in  the  cold  state.  The  cold  drawing  or  rolling 
changes  the  structure  of  the  meta],  increasing  its  strength  and 
brittleness,  and  consequently  the  original  ductility  must  be 
restored  by  an  annealing  operation. 

48.  Bronze. — Bronze  is  an  alloy  of  copper  and  tin.     Zinc  is 
sometimes  added  to  cheapen  the  alloy,  or  to  change  its  color  and 
to  increase  its  malleability. 


40  BRONZE  [CHAP.  II 

(a)  Commercial  bronze. — Commercial   bronze  is  acid-resisting 
and  contains  90  per  cent,  of  copper  and  10  per  cent,  of  tin.     This 
metal  has  been  used  successfully  for  pump  bodies,  also  for  thrust 
collars  subjected  to  fairly  high  pressures.     Another  bronze  which 
has  proven  very  serviceable  for  gears  and  worm  wheels  where 
noiseless  operation  is  desired,  contains  89  per  cent,  of  copper  and 
11  per  cent,   of  tin.     A  form  of  bronze  known  as  gun  metal 
has  the  following  approximate  composition :  88  per  cent,  of  copper; 
10  per  cent,  of  tin;  and  2  per  cent,  of  zinc.     It  is  used  for  high- 
grade  bearings  subjected  to  high  pressures  and  high  speeds. 

(b)  Phosphor  bronze. — Phosphor  bronze  varies    somewhat  in 
composition,  but  in  general  is  about  as  follows:  80  per  cent, 
copper;  10  per  cent,  tin;  9  per  cent,  lead;  and  1  per  cent,  phos- 
phorus.    It  is  easily  cast  and  is  as  strong  or  stronger  in  tension 
than  cast  iron.     It  is  a  very  serviceable  bearing  metal  and  is 
used  for  bearings  subjected  to  heavy  pressures  and  high  speeds; 
for  example,  locomotive  cross-head  bearings,  crankpin  bearings, 
and  bearings  on  grinders  and  blowers. 

A  phosphor  bronze  intended  for  rolling  into  sheets  or  drawing 
into  wire  contains  about  96  per  cent,  of  copper,  4  per  cent,  of  tin, 
and  sufficient  phosphorus  to  deoxidize  the  mixture.  The  tensile 
strength  of  such  a  phosphor  bronze  is  equal  to  that  of  steel. 

(c)  Manganese  bronze. — By  the  term  manganese  bronze,  as 
commonly  used,  is  meant  an  alloy  consisting  largely  of  copper  and 
zinc  with  small  percentages  of  other  elements  such  as  aluminum, 
tin  and  iron.     In  reality  many  of  the  so-called  manganese  bronzes 
are  not  bronzes  at  all,  but  brasses;  however,  there  are  several 
compositions  in  use  in  which  the  proportion  of  zinc  is  small 
compared  to  the  amount  of  tin  and  these  are,  strictly  speaking, 
bronzes.     Many  of  the  commercial  manganese  bronzes  contain 
no  manganese  whatever,  the  latter  being  used  merely  as  a  de- 
oxidizing agent. 

Due  to  its  high  tensile  strength  and  ductility,  manganese 
bronze  is  well  adapted  for  castings  where  great  strength  and 
toughness  are  required.  The  hubs  and  blades  of  propellers  and 
certain  castings  used  in  automobile  construction  are  frequently 
made  of  this  alloy.  It  is  not  nearly  as1  satisfactory  as  phosphor 
bronze  when  used  for  bearings.  A  manganese  bronze  made  of 
56  per  cent,  of  copper,  43.5  per  cent,  zinc  and  0.5  per  cent, 
aluminum  possesses  high  tensile  strength  and  is  suitable  for  the 


ART.  49]  ALUMINUM  41 

service  just  mentioned.     Manganese  bronze  may  also  be  rolled 
into  sheets  or  bars,  or  drawn  into  wire. 

(d)  Aluminum  bronze. — Aluminum  bronze  is  formed  by  adding 
not  to  exceed  11  per  cent,  of  aluminum  to  copper,  thus  producing 
an  alloy  having  great  strength  and  toughness.  An  alloy  con- 
taining 90  per  cent,  of  copper  and  10  per  cent,  aluminum  with  a 
trace  of  titanium  has  given  very  satisfactory  service  when  used 
for  machine  parts  requiring  strength  and  toughness,  and  at  the 
same  time  subject  to  wear;  for  example,  a  worm  wheel.  The 
last-named  composition  produces  an  alloy  that  has  an  ultimate 
tensile  strength  equal  to  that  of  a  medium-carbon  steel.  Ac- 
cording to  tests  made  at  Cornell  University,  the  coefficient  of 
friction  of  this  type  of  aluminum  bronze  is  0.0018,  thus  making 
it  suitable  for  bearings,  and  experience  has  shown  that  for  accu- 
rately fitted  bearings,  the  results  are  very  satisfactory.  The 
titanium  in  the  above  composition  is  added  to  insure  good  solid 
castings.  In  addition  to  the  uses  mentioned  above,  this  type 
of  bronze,  due  to  its  ability  to  resist  corrosion,  may  be  used  for 
parts  exposed  to  the  action  of  salt  water,  tanning  and  sulphite 
liquids. 

49.  Monel  Metal. — Monel  metal  is  a  combination  of  approxi- 
mately 28  per  cent,  of  copper,  67  per  cent,  of  nickel  and  small  per- 
centages of  manganese  and  iron.     It  has  a  high  tensile  strength, 
is  ductile,  and  has  the  ability  to  resist  corrosion.     It  may  be  used 
to  produce  castings  having  an  ultimate  strength  of  65,000  pounds 
per  square  inch.  When  used  for  rolling  into  sheets  or  bars,  the 
strength  is  increased  from  25  to  40  per  cent. 

Monel  metal  presents  no  difficulties  in  machining,  nor  in 
forging  operations  if  worked  quickly.  Like  copper,  it  is  impos- 
sible to  weld  it  under  the  hammer,  but  it  can  be  welded  by  means 
of  the  oxy-acetylene  flame  or  by  electricity.  Since  this  alloy  is 
non-corrodible  it  is  used  largely  for  propeller  blades,  pump  rods, 
high-pressure  valves,  and  steam-turbine  blading. 

50.  Aluminum. — Within  the  last  few  years  aluminum  alloys 
have  been  used  rather  extensively  for  many  different  machine 
parts.     Pure  aluminum  is  very  ductile  and  may  be  rolled  into 
very  thin  plates  or  drawn  into  fine  wire.     It  may  also  be  cast, 
but  the  casting  produced  has  a  coarse  texture  and  for  that  reason 
pure  aluminum  is  used  but  little  for  castings.     For  good  com- 
mercial casting,  aluminum  alloys  are  used.     The  alloys  recom- 
mended by  the  Society  of  Automobile  Engineers  are  the  following : 


42  BABBITT  METAL  [CHAP.  II 

(a)  Aluminum  copper. — The  aluminum  copper  alloy  contains 
not  less  than  90  per  cent,  of  aluminum,  7  to  8  per  cent,  of  copper, 
and  the  impurities  consisting  of  carbon,  iron,  silicon,  manganese 
and  zinc  shall  not  exceed  1.7  per  cent.  This  is  a  very  light 
material;  is  tough,  possesses  a  high  degree  of  strength,  and  may 
be  used  for  castings  subjected  to  moderate  shocks. 

(6)  Aluminum-copper-zinc. — An  alloy,  made  of  not  less  than 
80  per  cent,  of  aluminum,  from  2  to  3  per  cent,  of  copper,  not 
more  than  15  per  cent,  of  zinc,  and  not  to  exceed  0.40  per  cent,  of 
manganese,  gives  a  light-weight,  close-grained  material  that  can 
be  cast  easily  and  will  be  free  from  blow  holes.  The  castings  pro- 
duced are  very  strong  and  are  capable  of  resisting  moderate 
shocks. 

(c)  Aluminum  zinc. — The  alloy  containing  65  per  cent,  alu- 
minum and  35  per  cent,  zinc  is  intended  for  castings  subjected  to 
light  loads.  It  is  quite  brittle  and  is  used  for  footboards  and 
other  similar  parts  of  an  automobile.  It  is  about  the  cheapest 
aluminum  alloy  that  is  now  in  use. 

Due  to  the  excessive  shrinkage  to  which  all  aluminum  castings 
are  subjected,  great  care  must  be  exercised  in  their  design.  Thick 
sections  should  never  join  thin  sections  on  account  of  cracks  that 
are  very  likely  to  show  up  in  the  finished  castings.  In  order  to 
obtain  the  best  results,  all  parts  should  be  given  as  nearly  a  con- 
stant or  uniform  section  as  is  practical,  and  strength  combined 
with  light  weight  may  be  obtained  by  proper  ribbing. 

Cast  aluminum  is  used  successfully  in  the  construction  of  the 
framework  of  automobile  motors,  thus  saving  in  weight  and  aid- 
ing in  cooling  the  motor.  It  is  also  used  for  the  construction  of 
gear  cases,  pistons  and  clutch  parts.  For  machine  tools  such  as 
planers,  pulleys  are  frequently  made  of  cast  aluminum  so  as  to 
decrease  the  inertia  of  the  rotating  parts.  The  bodies  or  frame- 
work of  jigs  are  occasionally  made  of  cast  aluminum,  thus  mak- 
ing them  easier  to  handle. 

61.  Babbitt  Metal. — The  term  babbitt  metal  generally  refers  to 
an  alloy  consisting  of  copper,  tin  and  zinc  or  antimony  and  in 
which  the  tin  content  exceeds  50  per  cent. 

(a)  Genuine  babbitt  metal. — The  alloy  containing  copper,  tin, 
and  antimony  is  usually  called  genuine  babbitt  metal.  Accord- 
ing to  the  Society  of  Automobile  Engineers,  the  follownig  specifi- 
cations will  produce  a  high  grade  of  babbitt  that  should  give  ex- 
cellent results  when  used  for  such  service  as  connecting-rod 


ART.  52]  WHITE  BRASS  43 

bearings,  automobile  motor  bearings,  or  any  other  machine 
bearings  subjected  to  similar  service:  copper,  7  per  cent.;  anti- 
mony, 9  per  cent. ;  tin,  84  per  cent.  There  are  a  large  number  of 
commercial  grades  of  babbitt  metals,  many  of  which  have  a 
high  percentage  of  lead  and  consequently  sell  at  a  low  price. 

(b)  White  brass. — The  alloy  called  white  brass  is  in  reality  a 
babbitt  metal,  since  its  tin  content  exceeds  50  per  cent.,  as  the 
following  specifications  adopted  by  the  Society  of  Automobile 
Engineers  show:  3  to  6  per  cent,  of  copper;  28  to  30  per  cent,  of 
zinc;  and  not  less  than  65  per  cent,  of  tin.  This  alloy  is  recom- 
mended for  use  in  automobile  engine  bearings  and  generous  lubri- 
cation must  be  provided  to  get  the  best  results. 

Another  well-known  alloy  of  this  type  is  that  known  as  Parsons 
white  brass,  containing  2.25  per  cent,  of  copper,  64.7  per  cent,  of 
tin,  32.9  per  cent,  of  zinc  and  0.15  per  cent,  of  lead.  It  gives 
excellent  service  in  bearings  subjected  to  heavy  pressures  such  as 
are  found  in  marine  and  stationary  engine  practice;  also  in  con- 
nection with  high-speed  service  such  as  prevails  in  saw-mill 
machinery,  paper  and  pulp  machinery  and  in  electric  generators. 

As  a  rule,  white  brass  is  hard  and  tough,  and  to  get  the  best 
results  it  must  be  poured  at  a  very  high  temperature,  and  should 
then  be  peened  or  hammered  all  over  before  machining  the 
bearing. 

HEAT  TREATMENTS 

52.  Heat-treating  Processes. — The  term  heat  treatment  is 
applied  to  all  processes  of  heating  and  cooling  steel  through  cer- 
tain temperature  ranges  in  order  to  improve  the  structure,  and 
at  the  same  time  produce  certain  definite  and  desired  character- 
istics. The  processes  involved  in  heat  treatments  are  as  follows: 

(a)  Annealing. — The  object  of  annealing  steel  is  to  remove  the 
internal  stresses  due  to  cooling  as  well  as  to  produce  a  finer  tex- 
ture in  the  material.     In  general,  annealing  reduces  hardness  and 
increases  the  tensile  strength  and  elongation  of  the  steel. 

(b)  Hardening. — Steel  is  hardened  so  as  to  produce  a  good 
wearing  surface  or  a  good  cutting  edge.     The  effect  of  hardening 
is  to  raise  the  elastic  limit  and  the  ultimate  strength  of  the  steel 
and  at  the  same  time  reduce  its  ductility.     When  the  carbon  con- 
tent of  the  steel  is  0.5  per  cent,  or  over,  the  metal  becomes  brittle 
due  to  the  stresses  induced  by  the  sudden  quenching. 

(c)  Tempering. — The  process  of  tempering  consists  of  reheating 


44  HEAT  TREATMENTS  [CHAP.  II 

the  hardened  steel  in  order  to  restore  some  of  the  ductility  and 
softness  lost  in  the  hardening  process.  This  means  that  the  elas- 
ticity and  tensile  strength  are  reduced  below  the  values  for  the 
hardened  steel,  but  are  higher  than  those  prevailing  in  the  original 
material. 

(d)  Case-hardening. — By  the  process  of  case-hardening,  the 
outer  shell  or  skin  of  a  piece  of  steel  is  converted  into  a  high- 
carbon  steel  while  the  material  on  the  inside  remains  practically 
unchanged.  The  type  of  steel  to  which  this  process  is  applied 
generally  has  a  carbon  content  of  0.10  to  0.20  per  cent.,  and  ordi- 
narily should  not  contain  more  than  0.25  per  cent,  manganese  or 
the  case  produced  becomes  too  brittle.  Case-hardening  may 
also  be  applied  to  nickel  steel,  chrome  steel,  chrome-nickel  steel, 
or  chrome-vanadium  steel. 

To  case-harden,  the  pieces  are  packed  in  carbonaceous  material 
in  special  boxes  that  are  air-tight.  These  boxes  with  their  con- 
tents are  then  placed  in  a  furnace  in  which  the  temperature  is 
brought  up  to  about  1,500°F.,  and  maintained  at  that  tempera- 
ture for  a  definite  time  so  as  to  produce  the  desired  result.  The 
pieces,  after  receiving  this  first  treatment,  may  be  quenched  and 
are  ready  for  use.  In  order  to  get  better  results,  the  boxes  and 
their  contents  are  allowed  to  cool  in  the  furnace  or  in  the  air 
to  about  1,200°,  and  are  then  again  subjected  to  a  high  tem- 
perature after  which  the  contents  are  quenched.  A  few  of  the 
materials  used  for  packing  the  steel  in  the  boxes  are  as  follows: 
crushed  bone;  charred  leather;  barium  carbonate  and  charcoal; 
wood  charcoal  and  bone  charcoal. 

The  above  method  requires  considerable  time,  and  very  often 
it  is  desirable  to  produce  quickly  a  case-hardening  effect  which 
need  not  penetrate  the  material  very  far.  This  may  be  accom- 
plished by  the  use  of  a  mixture  of  powdered  potassium  cyanide 
and  potassium  ferrocyanide,  or  a  mixture  of  potassium  ferro- 
cyanide  and  potassium  bichromate.  In  general,  case-hardening 
is  used  when  a  machine  part  must  have  a  very  hard  surface  in 
order  to  resist  wear  or  impact,  a>nd  when  the  interior  of  the  piece 
must  be  tough  so  as  to  resist  fracture. 

63.  S.  A.  E.  Heat  Treatments. — In  January,  1912,  the  Society 
of  Automobile  Engineers  adopted  a  series  of  so-called  heat  treat- 
ments which  they  recommend  for  use  with  the  various  types  of 
steel  employed  in  automobile  construction.  Each  heat  treat- 
ment is  designated  by  a  letter  and  at  the  present  time  seventeen 


ART.  53]  HEAT  TREATMENTS  45 

different  treatments  are  included  in  the  above-mentioned  list. 
The  specifications  are  complete  as  may  be  seen  from  the  follow- 
ing, taken  from  the  Report  of  the  Iron  and  Steel  Division  of  the 
Standards  Committee  of  the  above-mentioned  society. 

Treatment  A. — For  screws,  pins  and  other  similar  parts 
made  from  0.15  to  0.25  per  cent,  carbon  steel,  and  for  which 
hardness  is  the  only  requirement,  the  simple  form  of  case-harden- 
ing designated  as  Heat  Treatment  A,  will  answer  very  well.  After 
the  piece  has  been  forged  or  machined,  treat  it  as  follows : 

1.  Carbonize  at  a  temperature  between  1,600°  and  1,750°F. 

2.  Cool  slowly  or  quench. 

3.  Reheat  to  1,450°  to  1,500°F.  and  quench. 

Treatment  C. — Steel  containing  from  0.25  to  0.35  per  cent, 
carbon  is  used  for  axle  forgings,  driving  shafts,  and  other  struc- 
tural parts,  and  in  order  to  get  better  service  from  this  grade 
of  steel,  the  parts  after  being  forged  or  machined  should  be  heat- 
treated,  the  simplest  form  of  which  is  given  by  the  following 
specifications: 

1.  Heat  to  1,475°  to  1,525°F. 

2.  Quench. 

3.  Reheat  to  600°  to  1,200°F.  and  cool  slowly. 

In  the  third  operation,  namely  that  of  drawing,  each  piece  must 
be  treated  individually;  for  example,  if  considerable  toughness 
with  no  increase  in  strength  is  desired,  the  upper  drawing  tem- 
peratures must  be  used;  while  with  parts  that  require  increased 
strength  and  little  toughness,  the  lower  temperatures  will 
answer. 

Treatment  K. — Treatment  K,  specifications  for  which  are 
given  below,  is  applicable  to  parts  made  of  nickel  and  nickel- 
chromium  steels,  in  which  extremely  good  structural  qualities  are 
desired : 

1.  Heat  to  1,500°  to  1,550°F. 

2.  Quench. 

3.  Reheat  to  1,300°  to  1,400°F. 

4.  Quench. 

5.  Heat  to  600°  to  1,200°F.  and  cool  slowly. 

In  reality  this  is  a  double  heat  treatment,  which  produces  a 
finer  structure  of  the  material  than  is  possible  with  only  one 
treatment. 

Treatment  V.: — Springs  made  from  silicon-manganese  steel 
are  treated  as  follows: 


46  GALVANIZING  [CHAP.  II 

1.  Heat  to  1,650°  to  1,750°F. 

2.  Quench. 

3.  Reheat  to  a  temperature  between  600°  and  1,400°F.,  and 
cool  slowly. 

PREVENTION  OF  CORROSION 

To  prevent  corrosion  of  iron  and  steel  it  is  necessary  to  protect 
the  surfaces  by  means  of  some  form  of  coating  which  may  be 
either  of  a  non-metallic  or  metallic  nature.  In  the  non-metallic 
method,  the  parts  are  coated  with  a  paint,  enamel  or  varnish,  the 
efficiency  of  which  depends  on  its  being  more  or  less  air-tight. 
This  method  is  far  from  satisfactory  due  to  the  chemical  changes 
causing  the  coating  to  peel  off  or  to  become  porous.  In  the 
metallic  method  the  parts  are  coated  with  some  other  metal, 
generally  zinc,  though  sometimes  copper  or  aluminum,  is  used. 
There  are  three  distinct  processes  of  putting  a  zinc  coating  on 
iron  and  steel,  as  follows :  hot-galvanizing,  electro-galvanizing,  and 
shererdizing. 

54.  Galvanizing. — (a)  Hot-galvanizing  consists  in  dipping  the 
parts,  which  have  been  cleaned  previously,  into  molten  spelter 
having  a  temperature  of  from  700°  to  900°F.  To  cause  the 
spelter  to  adhere  to  the  surfaces  of  the  articles,  a  soldering  flux 
(metallic  chlorides)  is  used.  The  zinc  deposited  on  the  parts  is 
not  chemically  pure,  and  the  impurities  increase  with  continued 
use  of  the  molten  spelter.  Due  to  these  impurities,  the  coating 
is  more  or  less  brittle  and  will  crack  easily.  The  thickness  of  the 
coating  is  far  from  uniform.  The  process  just  described  is  the 
oldest  known  for  coating  iron  and  steel  with  zinc. 

(b)  Electro-galvanizing. — This  process  is  also  known  as  cold- 
galvanizing  and  consists  in  depositing  zinc  on  the  parts,  previously 
cleaned,  by  means  of  electrolysis.  By  this  method  any  size  of 
article  may  be  treated,  and  it  is  claimed  that  the  deposit  consists 
of  chemically  pure  zinc.  The  thickness  of  the  coating  is  more 
easily  controlled  by  this  process  than  by  the  one  discussed  in  the 
previous  paragraph. 

65.  Shererdizing. — In  the  process  known  as  shererdizing,  the 
articles,  after  they  are  cleaned,  are  packed  with  zinc  dust  in  an 
air-tight  drum.  To  prevent  the  oxidation  of  the  zinc  by  the  air 
inside  of  the  drum,  a  small  amount  of  pulverized  charcoal  is 
mixed  with  the  contents  of  the  drum.  The  drum,  after  being 


ART.  55]  SHERERDIZING  47 

sealed,  is  placed  into  a  specially  constructed  oven  in  which  it  is 
brought  up  to  a  temperature  approximately  200°  below  the  melt- 
ing point  of  zinc.  To  get  an  even  distribution  of  the  heat  and 
at  the  same  time  to  produce  an  even  coating  on  the  articles,  the 
drum  is  rotated  continually.  By  means  of  this  process  it  is 
possible  to  produce  a  homogeneous  deposit  of  zinc,  the  thick- 
ness of  which  depends  upon  the  length  of  time  the  articles  are 
allowed  to  remain  in  the  oven. 


CHAPTER  III 

FASTENINGS 
RIVETS  AND  RIVETED  JOINTS 

66.  Rivets. — The  most  common  method  of  uniting  plates,  as 
used  in  boilers,  tanks  and  structural  work,  is  by  means  of  rivets. 
A  rivet  is  a  round  bar  consisting  of  an  upset  end  called  the  head, 
and  a  long  part  called  the  shank.  It  is  a  permanent  fastening, 
removable  only  by  chipping  off  the  head.  Rivets  should  in 
general  be  placed  at  right  angles  to  the  forces  tending  to  cause 
them  to  fail,  and  consequently  the  greatest  stress  induced  in  them 
is  either  that  of  shearing  or  of  crushing.  If  rivets  are  to  resist 
a  tensile  stress,  a  greater  number  should  be  used  than  when  they 
are  to  resist  a  shearing  or  a  crushing  stress. 

Rivets  are  made  of  wrought  iron,  soft  steel,  and  nickel  steel. 
They  are  formed  in  suitable  dies  while  hot  from  round  bars  cut 
to  proper  length.  The  shank  is  usually  cylindrical  for  about  one- 
half  its  length,  the  remaining  portion  tapering  very  slightly. 
In  applying  rivets,  they  are  brought  up  to  a  red  heat,  placed  in 
the  holes  of  the  plates  to  be  connected,  and  a  second  head  is 
formed  either  by  hand  or  machine  work.  Generally  speaking, 
machine  riveting  is  better  than  hand  work,  as  the  hole  in  the 
plates  is  nearly  always  filled  completely  with  the  rivet  body, 
while  in  hand  work,  the  effect  of  the  hammer  blow  does  not 
appear  to  reach  the  interior  of  the  rivet,  and  produce  a  move- 
ment of  the  metal  into  the  rivet  hole. 

57.  Rivet  Holes. — For  the  sake  of  economy  rivet  holes  are 
usually  punched.  There  are  two  serious  objections  to  thus 
forming  the  holes.  The  metal  around  the  holes  is  injured  by  the 
lateral  flow  of  the  metal  under  the  punch;  however,  this  objection 
may  be  obviated  by  punching  smaller  holes  and  then  reaming 
them  to  size.  Secondly,  the  spacing  of  the  holes  in  the  parts  to 
be  connected  is  not  always  accurate  in  the  case  of  punching,  so 
it  becomes  necessary  to  ream  out  the  holes,  in  which  case  the 
rivets  may  not  completely  fill  the  holes  thus  enlarged,  or  to  use 
a  drift  pin.  The  drift  pin  should  be  used  only  with  a  light- 
weight hammer. 

48 


ART.  58] 


FORMS  OF  RIVETS 


49 


The  diameter  of  the  rivet  hole  is  about  KG  inch  larger  than 
that  of  the  rivet.  This  rule  is  subject  to  some  variation,  depend- 
ing upon  the  class  and  character  of  the  work.  The  clearance 
given  the  rivet  allows  for  some  inaccuracy  in  punching  the  plate 
and  in  addition  permits  driving  the  rivet  when  hot.  Drilling 
the  holes  is  the  best  method  for  perforating  the  plates.  The  late 
improvements  in  drilling  machinery  have  made  it  possible  to 
accomplish  this  work  with  almost  the  same  economy  as  in  punch- 
ing. The  metal  is  not  injured  by  the  drilling  of  holes;  indeed, 
there  are  tests  which  show  an  increase  in  the  strength  of  the  metal 
between  the  rivet  holes. 

58.  Forms  of  Rivets. — Rivets  are  made  of  a  very  tough  and 
ductile  quality  of  iron  or  steel.  They  are  formed  in  dies  from 


* — 3d  "1 


row- 


(C) 


the  round  bar  while  hot,  and  in  this  condition  are  called  rivet 
blanks.  For  convenience,  the  head  which  is  formed  during  the 
process  of  driving  is  called  the  point,  to  distinguish  it  from  the 
head  that  is  formed  in  making  the  rivet  blank.  The  amount  of 
shank  necessary  to  form  the  point  depends  upon  the  diameter 
of  the  rivet.  Since  the  length  of  the  rivet  is  measured  under  the 
head,  the  length  required  is  equal  to  the  length  of  shank  neces- 
sary to  form  the  point  plus  the  grip  or  thickness  of  plates  to  be 
riveted  together.  The  various  forms  of  rivet  points  and  their 
proportions,  as  used  in  riveted  joints,  are  illustrated  in  Fig.  7. 
In  addition  to  the  proportions  for  the  points,  the  figure  also  gives 
the  length  of  shank  required  to  form  these  points.  The  style 
of  point  shown  in  Fig.  7  (a)  is  called  the  steeple  point;  that  illus- 
trated by  Fig.  7(6)  is  known  as  the  button  point ,  while  the  counter- 
sunk point  is  represented  by  Fig.  7(c).  The  lengths  of  rivets 


50  FORMS  OF  HEADS  [CHAP.  Ill 

should  always  be  taken  in  quarter-inch  increments  on  account  of 
stock  sizes.  Any  length  up  to  five  or  six  inches,  however,  may 
be  obtained,  but  the  odd  §izes  will  cost  more  than  the  standard 
sizes. 

59.  Forms  of  Heads. — Rivets  with  many  different  forms  of 
heads  may  be  found  in  mechanical  work,  but  the  ones  in  general 
use  in  boiler  work  are  only  three,  namely,  cone  head,  button  head 
and  countersunk  head.  These  are  shown  in  Fig.  7 (a),  (6),  and 
(c),  respectively.  The  proportions  advocated  by  different  manu- 
facturers vary  somewhat;  those  given  in  Fig.  7  are  used  by  the 
Champion  Rivet  Company.  The  steeple  point,  Fig.  7 (a),  is  one 
easily  made  by  hand  driving  and  is  therefore  much  used.  This 
form,  however,  is  weak  to  resist  tension  and  should  not  be  used 
on  important  work. 

The  cone  head,  Fig.  7 (a)  is  one  of  great  strength  and  is  used  a 
great  deal  in  boiler  work.  It  is  not  generally  used  as  a  form  for 
the  point  on  account  of  difficulty  in  driving.  The  button-head 
type,  Fig.  7(6),  is  widely  used  for  points  and  may  be  easily  formed 
in  hand  driving  by  the  aid  of  a  snap.  The  countersunk  point 
weakens  the  plate  so  much  that  it  is  used  only  when  projecting 
heads  would  be  objectionable,  as  under  flanges  of  fittings.  Its 
use  is  sometimes  imperative  for  both  heads  and  points,  but  it 
should  be  avoided  whenever  possible.  The  countersink  in  the 
plate  should  never  exceed  three-fourths  of  the  thickness  of  the 
plate,  and  for  that  reason,  the  height  of  the  rivet  point  is  generally 
from  J{ e  to  %  inch  greater  than  the  depth  of  the  countersink. 
The  point  then  projects  by  that  amount,  or  if  the  plate  is  required 
to  be  perfectly  smooth,  the  point  is  chipped  off  level  with  the 
surface. 

RIVETED   CONNECTIONS 

There  are  three  general  groups  of  riveted  connections  or  joints: 
the  first  of  these  includes  all  types  of  joints  met  with  in  the 
construction  of  tanks  and  pressure  vessels;  the  second  group, 
commonly  called  structural  joints,  includes  those  that  are  com- 
mon to  cranes,  structures,  and  machinery  in  general;  the  third 
group  includes  those  joints  used  in  the  construction  of  the  hulls 
of  ships.  It  is  evident,  that  in  the  first  group,  in  addition  to 
forming  a  rigid  connection  between  two  or  more  members,  the 
joint  must  also  be  made  secure  against  leakage.  In  the  third 
group  mentioned,  strength,  stiffness  and  durability  are  the  im- 


ART.  60] 


TYPES  OF  JOINTS 


51 


portant  points  desired,  as  well  as  proof  against  leakage ;  however, 
due  to  the  low  pressures  the  question  of  leakage  presents  no 
serious  difficulties. 

60.  Types  of  Joints. — Generally  speaking,  the  following  ar- 
rangements used  in  connecting  plates  by  means  of  rivets  are 
equally  well  adapted  to  the  three  groups  of  connections  mentioned 
in  the  preceding  paragraph. 

(a)  Lap  joints. — By  a  lap  joint  is  meant  an  arrangement 
which  consists  of  overlapping  plates  held  together  by  one  or  more 
rows  of  rivets.  If  one  row  of  rivets  is  used  as  shown  in  Fig. 
8 (a),  the  arrangement  is  called  a  single-riveted  lap  joint,  and  with 
two  rows  as  represented  in  Fig.  9,  it  is  called  a  double-riveted 


lap  joint.  In  the  latter  form  of  joint,  the  rivets  may  be  arranged 
in  two  ways,  namely  staggered  as  shown  in  Fig.  9 (a),  or  the  so- 
called  chain  riveted,  illustrated  in  Fig.  9(6). 

It  is  apparent  that  a  load  producing  a  tensile  stress  in  a  lap 
joint  tends  to  distort  the  joint  so  that  the  two  connected  plates 
are  practically  in  the  same  plane,  thus  inducing  a  bending  stress 
in  the  plate  as  well  as  tensile  and  shearing  stresses  in  the  rivet. 
This  distortion  is  not  quite  so  marked  in  double-riveted  lap 
joints,  due  to  the  additional  stiffness  given  by  the  greater  width 
of  the  overlap. 

(b)  Butt  joints. — When  plates  butt  against  each  other  and  are 
joined  by  overlapping  plates  or  straps,  the  connection  is  called 
a  butt  joint.  Such  a  joint  may  have  one  plate  on  the  outside,  or 
one  on  the  outside  and  another  on  the  inside,  as  shown  in  Fig.  10. 
As  in  lap  joints,  the  rivets  may  be  grouped  in  one  or  more  rows 
on  each  side  of  the  joint,  and  in  either  the  chain  or  staggered 


52 


TYPES  OF  JOINTS 


[CHAP.  Ill 


P  *i 


u/ 


(a) 


FIQ.  9. 


P  •— i 


(b) 


FIQ.  10. 


ART.  61]  FAILURE  OF  JOINTS  53 

riveted  arrangement,  illustrated  by  Figs.  10  and  12,  respectively. 
Butt  joints  having  two  cover  plates  are  not  subjected  to  the  ex- 
cessive distortion  found  in  lap  joints,  though  poor  workmanship 
may  cause  a  small  bending  stress  in  the  plates  and  a  tension  on 
the  rivet. 

61.  Failure  of  Joints. — In  arriving  at  the  intensity  of  stress  in 
any  of  the  types  of  joints  discussed  in  Art.  60,  we  shall  assume 
that  the  unit  stress  is  uniform  over  the  area  of  the  resisting  sec- 
tion, which,  of  course,  is  not  absolutely  correct  for  joints  subjected 
to  bending  nor  for  those  containing  two  or  more  rows  of  rivets. 
Furthermore,  in  the  following  discussion  no  allowance  will  be 
made  for  the  additional  holding  power  of  riveted  joints  due  to  the 
friction  between  the  plates.  American  designers  pay  no  atten- 
tion to  this,  as  experiments  made  at  the  United  States  Arsenal  at 
Watertown  seem  to  indicate  that  the  joints  will  slip  a  slight 
amount  at  loads  considerably  less  than  those  due  to  the  working 
pressures.  According  to  experiments  made  by  Bach,  the  fric- 
tional  resistance  of  a  riveted  joint  may  be  taken  approximately 
equal  to  15,000  pounds  per  square  inch  of  rivet  area. 

Experience  has  shown  that  riveted  joints  may  give  way  in  any 
one  of  the  following  ways : 

(a)  Shearing  of  the  rivet. — In  all  lap  joints  and  butt  joints  with 
one  strap,  the  rivets  tend  to  fail  along  one  section;  while  in  butt 
joints  with  two  straps,  failure  tends  to  take  place  along  two  sec- 
tions.    Thus  in  Fig.  8 (a),  the  tendency  would  be  for  the  rivet  to 
fail  along  the  line  where  the  plates  come  into  contact,  and  after 
failure,  the  condition  would  be  represented  by  Fig.  8(6).     Such  a 
rivet  is  said  to  be  in  single  shear,  and  in  case  two  sections  resist 
the  shearing  action,  the  rivet  is  in  double  shear.     If  P  represents 
the  force  transmitted  by  one  rivet,  and  d  the  diameter  of  the 
rivet  after  driving,  then 

P  =  ^^  (38) 

4 

(b)  Crushing  of  the  plate  or  the  rivet. — If  the  rivet  be  strong 
enough  to  resist  the  shearing  force,  the  plate  or  the  rivet  itself 
may  fail  by  crushing,  as  shown  at  A  in  Fig.  11.     The  force  upon 
the  rivet  is  distributed  over  a  semi-cylindrical  area  causing  a  dis- 
tribution of  pressure  upon  this  area  about  which  very  little  is 
known.     In  the  design  of  riveted  joints  it  is  customary  to  con- 
sider only  the  component  of  this  pressure  which  is  parallel  to  the 


54 


FAILURE  OF  JOINTS 


[CHAP.  Ill 


force  upon  the  rivet,  and  to  assume  that  it  is  distributed  over  the 
projected  area  of  the  rivet. 

The  unit  stress  indicated  by  this  crushing  action  is  called  a 
bearing  stress,  and  representing  it  by  Sb,  it  is  evident  that  the 
force  transmitted  by  one  rivet  is 


P  =  dtSb, 


(39) 


in  which  t  represents  the  thickness  of  the  plate.  From  (39),  it 
follows  that  for  any  particular  size  of  rivet  and  load  P,  the  bear- 
ing stress  depends  upon  the 
thickness  of  the  plate;  hence  it 
is  possible  to  have  different 
bearing  stresses  in  one  joint 
when  two  or  more  plates  of 
different  thicknesses  are  con- 
nected together. 

(c)  Tearing  of  the  plate. — In 
a  riveted  joint  subjected  to  a 
tension,  the  plates  may  be  pulled 
apart  along  the  line  of  rivets  as 
shown  at  B  in  Fig.  11.  Evidently 
the  least  area  of  the  plate  resist- 
ing this  tension  is  the  net  sec- 
tion between  consecutive  rivets. 
If  p  represents  the  pitch  of  the 
rivets,  then  the  force  transmitted  by  each  rivet  is 


FIG.   11. 


P  =  (p  -  d)tSt 


(40) 


(d)  Failure  of  the  margin. — By  the  term  margin,  also  called 
lap,  is  meant  the  distance  from  the  edge  of  the  plate  to  the  center 
of  the  line  of  rivets  nearest  the  edge,  as  shown  by  the  dimension 
a  in  Fig.  .11.  Failure  of  the  margin  may  occur  by  shearing  of  the 
plate  along  the  lines  in  front  of  the  rivet  as  shown  at  C  in  Fig.  11. 
With  actual  joints  in  use,  failure  in  this  way  is  not  likely  to  occur. 
It  follows  that  the  shearing  resistance  offered  by  the  plate  is 
2  atS, ;  hence,  the  force  each  rivet  is  capable  of  transmitting  is 


P  =  2atS, 


(41) 


The  margin  may  also  fail  by  tearing  open  as  shown  at  D  in 
Fig.  11.  This  failure  no  doubt  is  due  to  the  fact  that  in  a  joint 
subjected  to  tension,  the  material  in  front  of  the  rivet  behaves 


ART.  62]  BOILER  JOINT  55 

very  much  like  a  beam  loaded  at  the  center,  thus  causing  the 
plate  to  fail  by  breaking  open  on  the  tension  side,  usually  near  the 
center.  The  truth  of  the  above  statement  has  been  borne  out  by 
numerous  experiments.  A  rule  used  considerably  by  designers 
is  to  make  the  margin  never  less  than  one  and  one-half  times  the 
diameter  of  the  rivet,  and  experience  has  proven  that  joints  de- 
signed in  this  manner  seldom  fail  due  to  a  weak  margin.  The 
Association  of  the  Master  Steam  Boiler  Makers  recommends  that 
for  boiler  joints  the  margin  be  made  twice  the  diameter  of  the 
rivet.  This  marginal  distance  has  proven  very  satisfactory  in 
that  no  trouble  has  been  experienced  in  making  such  a  joint 
steam-tight  by  caulking. 

62.  Definitions. — In  the  investigation  of  the   stresses  in   a 
riveted  joint,  it  is  convenient  to  take  a  definite  length  of  the  joint 
as  the  basis  for  our  calculations.     This  length  may  or  may  not  be 
equal  to  the  pitch.     In  joints  having  two  or  more  rows  of  rivets, 
the  distance  between  the  rows  is  commonly  called  the  back  pitch, 
and  its  magnitude  is  approximately  70  per  cent,  of  the  pitch. 
An  examination  of  Figs.  10  and  12  shows  that  there  are  certain 
groups  or  arrangements  of  rivets  which  are  repeated  along  the 
entire  length  of  the  joint,  and  for  convenience  such  a  group  of 
rivets  may  be  called  a  repeating  group  and  the  length  occupied 
by  it  a  unit  length  of  a  riveted  joint.     In  the  analysis  of  any  type 
of  riveted  joint,  the  force  transmitted  by  such  a  repeating  group 
generally  forms  the  basis  of  all  calculations.     Another  term  used 
to  a  considerable  extent  in  connection  with  riveted  joints  is  the 
so-called  efficiency,  by  which  is  meant  the  ratio  that  the  strength 
of  a  unit  length  of  a  joint  bears  to  the  same  length  of  the  solid 
plate. 

RIVETED  JOINTS  IN  BOILER  CONSTRUCTION 

63.  Analysis  of  a  Boiler  Joint. — One  of  the  objects  desired  when 
designing  an  efficient  boiler  joint  is  to  make  the  joint  equally 
strong  against  failure  by  shearing,  bearing  and  tension ;  however, 
certain  modifications  are  necessary  for  economic  reasons  and,  as 
a  result,  the  actual  joint  as  finally  constructed  in  the  shop  will 
have  a  slightly  lower  efficiency  than  the  one  having  uniform 
strength.     In  order  to  illustrate  the  method  that  may  be  followed 
in  designing  a  joint  having  its  resistance  to  shearing,  bearing  and 
tension  approximately  the  same,  assume  the  double-riveted  lap 
joint  shown  in  Fig.  9.     From  this  figure  it  is  evident  that  the 


56  BOILER  JOINT  [CHAP.  Ill 

length  of  a  repeating  group  is  p,  the  pitch  of  the  rivets.  We 
shall  assume  that  the  two  plates  are  of  the  same  thickness  t, 
and  that  the  margin  was  made  of  sufficient  length  to  insure 
against  its  failure. 

The  resistance  P  due  to  the  shearing  of  the  rivets  in  a  unit 
length  of  the  joint  is 

P.--       .  (42) 


The  resistance  due  to  crushing  of  the  plate  and  the  rivets  is 

P  =  2  dtSb  (43) 

The  area  resisting  tension  is  (p  —  d)t,  and  multiplying  this  by 
the  unit  stress,  St,  the  total  resistance  against  tension  is 

•P  =  (p  -  d)tSt  (44) 

The  three  equations  just  determined  may  now  be  solved  simul- 
taneously if  it  is  desired  to  make  the  joint  of  equal  strength. 
Combining  (42)  and  (43),  we  obtain 

A          4tSb  fAtt 

d  =  7ST  (45) 

Equating  (42)  and  (44),  the  pitch  becomes 

P-1+jg  (46) 

Equating  (43)  and  (44),  it  follows  that 

p  =  d  +  ^  (47) 

&t 

Basing  the  size  of  the  rivet  upon  (45)  would  lead  to  odd  diame- 
ters that  are  not  obtainable,  since  the  commercial  sizes  vary 
by  J{  e-inch  increments  from  ^  inch  to  1%  inches  in  diameter. 
Hence,  with  the  use  of  commercial  sizes  of  rivets,  it  is  impossible 
to  make  the  joint  equally  strong  against  the  three  methods  of 
failure  discussed  above.  Furthermore  as  the  thickness  of  the 
plate  increases,  the  diameter  d  calculated  by  (45)  becomes  ex- 
cessively large,  thus  introducing  serious  difficulties  in  driving  such 
a  rivet.  Having  decided  upon  the  size  of  rivet,  the  pitch  may  be 
determined  by  means  of  (46)  and  (47),  but  it  may  be  necessary 
to  modify  the  calculated  pitch  so  as  to  insure  a  steam-tight  joint. 
From  this  discussion  it  is  apparent  that  the  group  of  theoretical 
formulas  derived  above  serves  merely  as  a  guide. 


ART.  64] 


EFFICIENCY  OF  BOILER  JOINTS 


57 


In  general,  the  method  of  procedure  to  be  used  in  designing 
riveted  joints  is  as  follows: 

(a)  Determine  expressions  for  the  various  methods  of  failure. 

(6)  Select  a  commercial  size  of  rivet,  so  that  it  may  be  driven 
readily. 

(c)  Having  selected  the  size  of  rivet,  determine  whether  the 
rivet  will  fail  by  shearing  or  by  crushing. 

(d)  Determine  the  pitch  by  equating  the  expression  for  the 
tearing  of  the  plate  to  that  giving  the  rivet  failure. 

(e)  Determine  the  probable  efficiency  of  the  joint. 

64.  Efficiency  of  the  Joint. — The  efficiency  of  a  riveted  joint  is 
defined  as  the  ratio  that  the  strength  of  a  unit  length  of  a  joint 
bears  to  the  same  length  of  the  solid  plate.  In  the  analysis  of 
the  double-riveted  lap  joint,  it  developed  that  there  were  three 
distinct  ways  that  the  joint  could  fail;  hence,  the  efficiency  of 
that  joint  depends  upon  the  expression  that  gives  the  minimum 
value  of  P.  In  a  double-riveted  butt  and  double-strap  joint, 
there  are  six  ways  that  failure  may  occur  and  whichever  is  the 
weakest  determines  the  probable  efficiency  of  the  joint. 

The  strength  of  the  solid  plate  of  thickness  t  and  unit  length 
L  is  t  LSt;  hence,  the  general  expression  for  the  efficiency  of  a 
riveted  joint  becomes 

minimum  P  , .  ^ 


E  = 


tLSt 


TABLE  9. — EFFICIENCY  OF  BOILER  JOINTS 


The  range  of  values  for  the  efficiency  E  for  the  various  types  of 
joints  used  in  boiler  design  is  given  in  Table  9.  These  values 
may  serve  as  a  guide 
in  making  assumptions 
that  are  necessary 
when  designing  joints 
for  a  particular  duty. 
In  case  the  actual  or 
calculated  efficiency 
does  not  agree  closely 
with  the  assumed 
value,  the  joint  will 
have  to  be  redesigned, 
until  a  fair  agreement 
is  obtained. 

65.  Allowable  Stresses. — In  order  to  design  joints  that  will 
give  satisfactory  service  in  actual  use,  considerable  attention  must 


Type  of  joint 

Efficiency 

Min. 

Max. 

Lap  joint. 

Single-riveted 
Double-riveted 
Triple-riveted 

45 

60 
65 

55 
70 
75 

85 

60 

75 

84 

Butt       joint 
with    two 
cover  plates 

Single-riveted 
Double-riveted 
Triple-riveted 
Quadruple-riveted 

65 
80 
88 
95 

58 


ALLOWABLE  STRESSES 


[CHAP.  Ill 


be  given  to  the  selection  of  the  proper  working  stresses  for  the 
materials  used.  At  the  annual  meeting  of  the  American  Society 
of  Mechanical  Engineers  held  in  December,  1914,  a  committee 
appointed  by  that  society  presented  an  extensive  report  in  which 
the  question  of  the  selection  of  the  material  is  discussed  very 
fully.  The  recommendations  are  as  follows: 


TABLE    10. — ULTIMATE  SHEARING 
STRESSES  IN  RIVETS 


TABLE   11. — THICKNESS    OF 

SHELL  AND  DOME  PLATES 

AFTER    FLANGING 


Kind  of 
rivet 

Ultimate  shearing 

Single 
shear 

Double 
shear 

Iron  
Steel  

38,000 
44,000 

76,000 
88,000 

Diameter  of  shell 

Minimum 
thickness 

36  and  under  .  .  . 
36  to  54  

H 

54  to  72  
72  and  over.  .  .  . 

H 
H 

TABLE    12. — THICKNESS    OF 
JOINT   COVER  PLATES 


BUTT 


(a)  In  the  calculations  for  steel  plates  when  the  actual  tensile 
strength  is  not  stamped  on  the  plates,  it  shall  be  assumed  as 
55,000  pounds  per  square  inch. 

(b)  The  ultimate  crushing  strength  of  steel  plate  shall  be  taken 
at  95,000  pounds  per  square  inch. 

(c)  In  rivet  calculations,  the  ultimate  shearing  strengths  given 

in  Table  10,  and  based  on  the 
cross-sectional  area  of  the 
rivet  after  driving,  shall  be 
used. 

(d)  To  obtain  the  allowable 
working  stresses,  the  ultimate 
strengths  given  above  must  be 
divided  by  the  so-called  factor 
of  safety,  the  value  of  which 
should  never  be  less  than  five. 

66.  Minimum  Plate  Thick- 
ness.— According  to  recom- 
mendations made  by  the  Boiler 
Code  Committee  of  the  American  Society  of  Mechanical  Engi- 
neers, no  boiler  plate  subjected  to  pressure  should  be  made  less 
than  Y±  inch  thick,  and  the  thicknesses  given  in  Table  11  for 
various  shell  diameters  may  serve  as  a  guide  in  designing  work. 


Thickness  of 
shell  plates 

Thickness  of 
cover  plates 

K  to  i^ 

%and  i 
KG  and 
Hto^e 
%  and  2' 

2  inclusive  

KG 

inclusive  
/ 

1  and  \y 

/ 

3  

ART.  67] 


SIZE  OF  RIVET  HOLES 


59 


For  the  thicknesses  of  the  cover  plate  for  butt  joints,  the  recom- 
mendations of  this  committee  are  given  in  Table  12. 

TABLE  13. — RECOMMENDED  SIZE  OF  RIVET  HOLES 


Plate 
thickness 

Diameter  of  rivet  holes 

Lap  joint 

Double-strap  butt  joint 

Single- 
riveted 

Double- 
riveted 

Triple- 
riveted 

Double- 
riveted 

Triple- 
riveted 

Quadruple- 
riveted 

K                  % 

lHe 

KG 

%2                % 

% 

% 

KG 

« 

lHe 

H 

% 

% 

« 

11/16 

3/8 

^6 

% 

lHe 

H 

l3/32 

15A6 

Ys 

KG 

1 

JK6 

7/8 

% 

lHe 

l5/32 

IMG 

15/16 

lMe 

X 

1 

lMe 

lHe 

% 

KG 

5/s 

lHa 

llAe 

1%6 

3A 

1K* 

We 

y* 

JK6 

' 

!Ke 

I 

We 

1 

!Ke 

In  Table  13  are  given  the  diameters  of  rivet  holes  for  dif- 
ferent plate  thicknesses  and  various  types  of  joints,  as  determined 
from  a  study  of  actual  joints  used  in  the  construction  of  boilers 
and  pressure  tanks. 

67.  Design  of  a  Boiler  Joint. — It  is  required  to  design  a  triple- 
riveted  double-strap  butt  joint  for  the  longitudinal  seam  of  a 
boiler  66  inches  in  diameter,  assuming  the  working  pressure  as 


60  DESIGN  OF  A  BOILER  JOINT  [CHAP.  Ill 

150  pounds  per  square  inch,  and  the  ultimate  tensile  strength  of 
the  plates  as  60,000  pounds  per  square  inch.  For  the  factor  of 
safety,  and  shearing  and  crushing  stresses,  use  the  values  recom- 
mended by  the  American  Society  of  Mechanical  Engineers. 

(a)  The  first  step  in  the  solution  of  this  problem  is  to  assume 
the  probable  efficiency  of  the  joint,  which  according  to  Table  9 
may  be  taken  as  85  per  cent. 

(6)  Determine  next  the  thickness  of  the  shell  plates  making 
proper  allowances  for  the  decrease  in  the  strength  of  the  shell 
due  to  the  joint.  The  formula  for  the  plate  thickness  is  de- 
termined by  considering  the  boiler  a  cylinder  with  thin  walls 
subjected  to  an  internal  pressure,  whence 

P'D  150  X  66 


2ESt   '-  2  X  0.85  X  12,000 


=  °'485  mch* 


Selecting  the  nearest  commercial  size,  the  thickness  of  the  shell 
plates  will  be  made  J/2  inch. 

(c)  The  cover  plates  or  straps  of  a  triple-riveted  butt  joint  for 
a  J^-inch  shell  should  be  %6  inch  thick,  according  to  Table  12, 
and  the  diameter  of  the  rivet  hole  as  given  in  Table  13  will  be 
iKe  inch,  thus  calling  for  1-inch  rivets. 

(d)  According  to  the  recommendations  of  the  American  Society 
of  Mechanical  Engineers,  the  following  ultimate  stresses  will  be 
used;£8  =  44,000  and  Sb  =  95,000,  from  which  the  following  unit 
values  are  obtained; 

-  =  39,000  pounds. 

dt'Sb  =  1^6  X  KG  X  95,000  =  44,150  pounds. 
=  ll/iQ  X  1A    X  95,000  =  50,470  pounds. 


(e)  Having  arrived  at  the  proper  plate  thicknesses  and  the 
diameter  of  the  rivets,  the  resistances  to  failure  of  the  joint  must 
be  investigated  in  order  to  establish  the  probable  pitch  of  the 
rivets.  A  triple-riveted  double-strap  butt  joint  similar  to  that 
shown  in  Fig.  12  may  fail  in  any  one  of  the  following  ways  : 

1.  Tearing  of  the  plate  between  the  rivet  holes  in  the  outer 
row.  —  Using  the  notation  prevailing  in  preceding  articles,  the 
magnitude  of  the  resistance  to  failure  by  tearing  of  the  plate 
between  the  rivet  holes  is 

P  =  (p-  d)tSt  (49) 


ART.  67] 


DESIGN  OF  A  BOILER  JOINT 


61 


2.  Tearing  of  the  plate  between  the  rivet  holes  in  the  second 
row,  combined  with  the  failure  of  the  rivet  in  the  outer  row. — 
An  inspection  of  Fig.  12  shows  that  before  the  plate  could  fail 
between  the  rivets  in  the  second  row,  the  rivet  in  the  outer  row 
would  have  to  fail  either  by  shearing  or  by  crushing,  hence  for 
this  case  two  separate  resistances  are  obtained  as  follows: 


(50) 
(51) 


P  =  (p  - 

P  =  (p  -2d)tSt 


FIG.  12. 

3.  Shearing  of  all  the  rivets. — It  is  evident  that  in  the  triple- 
riveted  butt  joint  shown  in  Fig.  12,  four  rivets  are  in  double  shear 
and  one  in  single  shear;  hence,  the  magnitude  of  the  resistance 
to  failure  is 

P  =  ?1*S.  (52) 


62  DESIGN  OF  A  BOILER  JOINT  [CHAP.  Ill 

4.  Crushing  of  all  the  rivets.  —  There  are  five  rivets  resisting 
crushing;  hence,  the  expression  for  the  resistance  to  crushing  is 

P  =  (4  1  +  t')dSb  (53) 

5.  Combined  crushing  and  shearing.  —  The  joint  may  also  fail 
by  the  crushing  of  the  four  rivets  on  the  inner  and  second  rows, 
and  the  shearing  of  the  rivet  in  the  outer  row;  hence,  the  com- 
bined resistances  of  these  rivets  is 

P  =  4  dtSb  +  ^-°  (54) 

4 

A  joint  of  the  type  discussed  above  should  be  designed  so  that 
the  strength  of  the  critical  sections  increases  as  these  sections 
approach  the  center  of  the  joint.  This  condition  is  fulfilled  by 
making  the  values  of  P  obtained  from  (50)  and  (51)  greater  than 
that  obtained  by  the  use  of  (49)  ;  that  is 


(p  -  2  d)tSt  +      p  >(p  -  d)tSt  (55) 

(p  -  2d)tSt  +  dt'Sb  >  (p  -  d)tSt  (56) 

From  (55)  it  follows  that  the  diameter  of  the  rivet  hole  becomes 

d>^->  (57) 

irSs 

and  simplifying  (56),  we  find  that 


The  expressions  given  by  (57)  and  (58)  must  be  satisfied,  if  it 
is  desired  to  make  the  triple-riveted  butt  joint  shown  in  Fig.  12 
stronger  along  the  inner  rows  than  at  the  outer  rows.  Having 
satisfied  these  equations  by  choosing  proper  values  for  d,  t  and 
t',  the  pitch  p  is  determined  by  equating  the  minimum  value  of 
P,  obtained  by  evaluating  (52),  (53)  and  (54),  to  that  obtained 
from  (49),  and  solving  for  p. 

Applying  the  principles  just  established  to  the  data  given 
above,  we  find  that  according  to  (57),  the  minimum  value  of  d 
is  0.87  inch,  and  from  (58)  the  minimum  value  of  t'  is  0.32  inch; 
hence  it  is  evident  that  the  values  assumed  above  will  insure 
increased  strength  of  the  joint  along  the  inner  rows. 

An  inspection  of  the  above  formulas  indicates  that  (54)  gives 


ART.  68]  RIVET  SPACING  63 

the  minimum  value  of  P,  and,  after  substitution,  we  find  that 
P  =  240,880  pounds.  Inserting  this  value  in  (49)  and  de- 
termining the  magnitude  of  the  pitch,  we  get  p  =  9.09  inches, 
say  9  inches. 

The  strength  of  the  solid  plate  is  9  X  M  X  60,000  =  270,000 
pounds;  hence,  from  (48),  the  efficiency 

E = IS  -  °-892  °r  89-2  p- cent- 

RIVETED   JOINTS  FOR  STRUCTURAL  WORK 

The  design  of  riveted  joints  for  structural  work  generally  calls 
for  the  selection  of  the  economical  size  of  the  members  required 
to  transmit  the  given  force,  in  addition  to  the  determination  of 
the  proper  size  and  number  of  rivets  to  be  used.  In  structural 
joints  the  size  of  the  rivet  depends  in  a  general  way  upon  the  size 
of  the  connected  members,  but  the  usual  sizes  are  %,  %  and  % 
inch  in  diameter.  Rivets  larger  than  %  inch  cannot  be  driven 
tight  by  hand  and  since  in  structural  work  many  of  the  joints 
must  be  put  together  in  the  field  by  hand  riveting,  it  is  evident 
that  %  inch  is  the  limiting  size  for  this  class  of  work.  Tables 
giving  the  maximum  size  of  rivets  that  can  be  used  with  the 
various  sizes  of  structural  shapes  may  be  found  in  the  hand  books 
published  by  the  several  steel  companies. 

68.  Rivet  Spacing. — In  the  spacing  of  rivets  the  following 
points  must  be  considered: 

(a)  If  rivets  are  spaced  too  closely,  the  material  between 
consecutive  rivets  may  be  injured  permanently. 

(6)  Too  close  spacing  might  interfere  with  the  proper  use  of  the 
snap  or  set  during  the  driving  operation. 

(c)  Rivets  that  are  spaced  far  apart  prevent  intimate  contact 
between  the  members;  water  and  dirt  may  collect  and  the  joint 
may  thus  deteriorate  by  rusting. 

(d)  Rivets  are  usually  spaced  according  to  rules  dictated  by 
successful  practice,  as  the  following  will  indicate.     The  minimum 
pitch  between  rivets  is  approximately  three  times  the  diameter 
of  the  rivet,  and  the  maximum  is  given  as  sixteen  times  the 
thickness  of  the  thinnest  plate  used  in  the  joint. 

(e)  For  gauge  lines  used  in  connection  with  the  various  struc- 
tural shapes,  the  steel  companies  hand  books  should  be  consulted. 


64  STRUCTURAL  JOINTS  [CHAP.  Ill 

69.  Types  of  Joints. — In  general,  it  may  be  said  that  the 
various  lap  and  butt  joints  used  in  structural  work  are  very 
similar  to  those  discussed  in  Art.  60.     In  addition  to  lap  and  butt 
joints,  there  are  a  great  variety  of  riveted  joints  in  which  the 
several  forms  of  structural  shapes  are  joined  together,  either 
with  or  without  the  use  of  connecting  plates  commonly  called 
gusset  plates.     Several  common  forms  of  such  joints  will  be 
discussed. 

The  following  order  of  calculations  is  common  to  practically 
all  structural  joints: 

(a)  From  the  magnitude  of  the  load  to  be  transmitted,  de- 
termine the  size  of  the  member. 

(b)  In  general  the  diameter  of  the  rivets  to  be  used  in  the 
connection  depends  upon  the  size  of  the  connected  members. 

(c)  Determine  the  number  of  rivets  required  in  each  member 
to  transmit  the  load  in  that  member.     This  number  depends 
upon  the  shearing  and  bearing  stresses,  whichever  determines 
the  method  of  failure. 

(d)  The  rivets  in  the  joint  must  be  arranged  or  spaced  in  such  a 
manner  that  in  the  case  of  a  tension  member  the  stress  along 
a  section  through  a  rivet  does  not  exceed  the  allowable  stress. 
To  determine  the  net  area  in  such  a  case  it  is  customary  to  con- 
sider the  size  of  the  rivet  hole  to  be  ^  inch  larger  than  the 
diameter  of  the  rivet.     For  compression  members,  the  area  of 
the  rivet  hole  is  never  considered  in  determining  the  net  area  of 
the  member. 

70.  Single  Angle  and  Plate.- — A  very  common  method  of  con- 
necting a  single  angle,  either  in  tension  or  compression,  to  a  plate 
is  shown  in  Fig.  13 (a).     It  is  apparent  that  the  connection  of  one 
leg  of  the  angle  to  the  gusset  plate  will  cause  the  angle  to  be 
loaded  eccentrically;  this  eccentricity  increases  the  stress  con- 
siderably over  that  due  to  central  loading.     The  determination 
of  the  additional  stress  due  to  the  moment  does  not  complicate 
the  problem  to  any  great  extent,  and  for  that  reason  the  analysis 
necessary  to  determine  the  size  of  the  angle  in  any  given  case 
should  be  made  as  complete  as  possible.     The  following  problem 
will  serve  to  illustrate  the  method  of  procedure  in  any  given  case. 

It  is  desired  to  determine  the  size  of  an  angle  and  the  number 
and  size  of  rivets  required  in  a  connection  similar  to  that  repre- 
sented in  Fig.  13,  in  which  the  force  P  acting  on  the  member 'e 
is  16,800  pounds.  Assume  the  allowable  stresses  in  tension, 


ART.  70] 


ANGLE  AND  PLATE 


65 


shearing  and  bearing  as  16,000,  10,000,  and  20,000  pounds  per 
square  inch,  respectively  and  the  thickness  of  the  gusset  plate 
as  K  inch. 

The  net  area  of  the  cross-section  of  the  required  angle,  assuming 


central  loading,  must  be 


1.05   sq.   in.     This  condition 


16,000 

would  be  met  by  a  3  by  2>^  by  J^-inch  angle  having  a  net  area 
of  1.09  square  inches  after  making  allowance  for  a  %-inch  rivet. 
Taking  account  of  the  eccentric  loading,  we  will  try  a  3^  by  3  by 


(b) 


FIG.  13. 

^£ -inch  angle,  having  a  gross  area  of  2.30  square  inches  and  a  net 
area  of  1.97  square  inches.  From  a  table  of  properties  of  struc- 
tural angles,  we  find  that  the  distance  Xi  in  Fig.  13(6)  is  0.83  inch, 
thus  making  the  eccentricity  of  the  load  P  equal  to  0.955  inch. 
Hence  applying  (17)  the  maximum  tensile  stress  in  the  angle  is 

16,430  pounds  per  square  inch 


16,800       16,800  X  0.95  X  0.83 
1.97  1.85 


which  is  assumed  as  sufficiently  close  to  the  allowable  stress  given 
above. 

To  obtain  the  number  of  rivets  in  the  joint,  determine  whether 
the  rivet  is  stronger  in  shear  or  in  bearing.  For  the  case  con- 
sidered, the  bearing  resistance  is  the  smaller,  having  a  value  of 
3,750  pounds  per  rivet;  hence  five  J^-inch  rivets  are  required. 

From  the  above  analysis,  it  is  apparent  that  the  stress  due  to 


66  BEAM  CONNECTIONS  [CHAP.  Ill 

the  eccentricity  of  the  load  P  cannot  be  disregarded,  and  further 
that  economy  of  material  is  obtained  by  loading  the  angle  cen- 
trally. The  latter  condition  is  considered  fulfilled  when  both  legs 
are  connected  to  the  gusset  plate.  Such  a  connection  is  effected  by 
the  use  of  a  clip  angle  g  as  shown  in  Fig.  13  (c),  provided  the  rivets 
are  divided  equally.  Assuming  that  the  joint  is  made  similar  to 
that  shown  in  Fig.  13  (c),  the  data  given  in  the  above  problem 
calls  for  a  3  by  2^  by  J^-inch  angle.  Each  angle  must  be  con- 
nected to  the  gusset  plate  by  means  of  three  rivets,  and  the  same 
number  must  be  used  for  connecting  together  the  two  angles. 
Tests  made  on  steel  angles  having  a  clip-angle  connection  with 
the  gusset  plate,  as  illustrated  in  Fig.  13  (c),  do  not  confirm  the 
analysis  just  given,  since  the  results  seem  to  indicate  that  very 
little  is  gained  by  the  use  of  such  angles. 

71.  End  Connections  for  Beams.  —  The  rivets  in  the  connec- 
tions used  on  the  ends  of  beams  are  subjected  to  a  secondary 
shearing  stress  in  addition  to  the  direct  stress  due  to  the  load  on 
the  joint,  as  the  following  analysis  will  show: 

According  to  the  steel  manufacturer's  handbook  the  standard 
connection  for  a  12  by  40-pound  I-beam  consists  of  two  6  by  4 
by  j^-inch  angles  7^  inches  long,  as  shown  in  Fig.  14  (a).  Fur- 
thermore, the  same  source  of  information  gives  8.2  feet  as  the 
minimum  length  of  span  for  which  the  connection  is  considered 
safe  when  used  with  a  beam  loaded  uniformly  to  its  full  capacity. 
The  uniform  load  that  the  beam  will  carry  without  exceeding 
a  fiber  stress  of  16,000  pounds  per  square  inch  is 

8  X  16,000  X  41.0 


82  x  12 


=  53,330  pounds 


This  gives  a  reaction  R  at  the  end  connection  of  26,665  pounds,  as 
shown  in  Fig.  14  (a).  It  is  evident  from  an  inspection  of  the 
figure  that  this  reaction  tends  to  rotate  the  connecting  angles 
about  the  center  of  gravity  of  the  rivet  group,  thus  causing  each 
rivet  to  be  subjected  to  a  shear  due  to  the  turning  moment,  in 
addition  to  the  direct  shear  caused  by  the  reaction. 

Due   to  the  reaction  R,  the  direct  shear  coming  upon  each 

O£l  A  A.  Px 

rivet  in  the  group  has  a  magnitude  of  -'-^  —  or  5,333  pounds. 

Due  to  the  turning  moment,  the  shearing  stress  produced  in  any 
rivet  in  the  group  is  proportional  to  the  distance  that  the  rivet  is 
from  the  center  of  gravity  of  the  group;  hence,  the  resisting  mo- 


ART.  71J 


BEAM  CONNECTIONS 


67 


ment  of  each  rivet  about  the  center  of  rotation  varies  as  the 
square  of  this  distance.  Letting  S'g  represent  the  secondary 
shear  in  the  rivet  nearest  to  the  center  of  gravity,  and  l\,  1%,  etc., 
the  distances  from  the  center  of  gravity  G  to  the  rivets  1,  2,  etc., 
respectively,  as  shown  in  Fig.  14(6),  then  the  external  moment 
M,  being  equal  to  the  summation  of  the  resisting  moments  due 
to  the  rivets,  is  given  by  the  following  expression : 


(59) 


"        l2x40lbIBeam 


FIG.  14. 

From  Fig.  14(6),  the  values  lh  l«,  etc.,  may  be  calculated,  and 
since  M  is  known,  the  magnitude  of  S'8  is  readily  obtained.  For 
the  data  at  hand  S'8  =  3,490  pounds;  hence,  the  shears  coming 
upon  the  various  rivets  are  as  follows: 

Secondary  shear  on  rivet  1  =  3,490  Ib. 
Secondary  shear  on  rivet  2  =  10,300  Ib. 
Secondary  shear  on  rivet  3  =  7,140  Ib. 
Secondary  shear  on  rivet  4  =  7,140  Ib. 
Secondary  shear  on  rivet  5  =  10,300  Ib. 

To  determine  the  resultant  shear  upon  each  rivet,  the  direct 
and  secondary  shears  must  be  combined.  This  may  be  done  by 


68  DOUBLE  ANGLE  AND  PLATE  [CHAP.  Ill 

algebraic  resolution,  or  graphically  as  shown  in  Fig.  14 (c).  It 
is  evident  that  rivets  2  and  5  are  subjected  to  the  heaviest  stress, 
the  magnitude  of  which  scaled  from  Fig.  14(c)  is  13,150  pounds; 
whence  the  unit  shearing  stress  in  each  of  these  %-inch  rivets  is 
14,880  pounds  per  square  inch.  Since  the  web  thickness  of  the 
12-inch  by  40-pound  beam  is  0.56  inch,  the  bearing  stress  coming 
upon  rivets  2  and  5  is  31,300  pounds  per  square  inch.  This 
problem  shows  the  importance  of  determining  the  actual  stresses 
in  the  rivets  of  eccentrically  riveted  connections. 

In  the  later  editions  of  the  steel  manufacturer's  hand  books,  it 
is  of  interest  to  note  that  the  "End  Connections  for  Beams  and 
Channels"  have  been  redesigned  and  for  the  size  of  beam  given 
in  the  preceding  problem  two  4  by  4  by  Jig-inch  angles  8^ 
inches  long  are  now  recommended  instead  of  those  mentioned 
above,  and  furthermore  only  three  %-inch  rivets  are  used. 

72.  Double  Angle  and  Plate. — A  form  of  connection  met  with 
occasionally  is  shown  in  Fig.  15.  It  is  desired  to  determine  the 
load  P  that  this  form  of  connection  will  safely  carry,  assuming  that 
all  rivets  are  %-inch  in  diameter  and  that  the  following  stresses 
shall  not  be  exceeded:  St  =  15,000;  Ss  =  10,000;  Sb  =  20,000. 

The  connection  may  fail  in  the  following  ways: 

(a)  The  rivets  in  the  outstanding  leg  of  the  lug  and  girder 
angles  may  fail  due  to  tension. 

(6)  The  rivets  may  shear  off  or  crush  in  the  vertical  legs  of  the 
lug  angle. 
•  (c)  The  rivets  may  sKear  off  or  crush  in  the  angles  A. 

(d)  The  lug  angles  may  fail  by  combined  tension  and  bending. 

The  specifications  for  structural  steel  work  do  not  recognize 
the  ability  of  rivets  to  resist  tension;  however,  for  secondary 
members  it  is  not  unusual  to  assume  the  permissible  stress  in 
rivets  subjected  to  tension  as  equivalent  to  the  permissible  shear- 
ing stress.  Upon  this  assumption,  the  eight  rivets  in  the  out- 
standing legs  of  the  lug  angles  are  capable  of  supporting  safely 
a  load  of  8  X  0.442  X  10,000  =  35,360  pounds.  From  the  de- 
tails shown  in  Fig.  15,  it  is  apparent  that  the  rivets  in  the  ver- 
tical legs  of  the  lug  angles  and  those  in  the  angles  A  are  of  equal 
strength,  hence  the  safe  load  that  they  are  capable  of  support- 
ing, as  measured  by  their  resistance  to  crushing,  is  3  X  M  X 
%  X  20,000  or  16,875  pounds. 

To  determine  the  bending  stress  in  the  lug  angles  it  is  assumed 
that  the  outstanding  legs  of  these  angles  are  equivalent  to  canti- 


ART.  72] 


DOUBLE  ANGLE  AND  PLATE 


69 


levers  having  the  load  applied  at  the  center  of  the  rivets.  Upon 
this  assumption,  the  maximum  bending  moment  occurs  in  the 

vertical  leg,  and  its  magnitude  in  this  case  is  determined  as 
p 

follows:  Let  TJ-T  be  the  vertical  load  coming  upon  each  inch  of 
2  Li 

length  of  the  lug  angle;  then  since  this  load  is  considered  as 
applied  at  the  center  of  the  rivet,  the  magnitude  of  the  bending 
moment  M  per  inch  of  length  of  the  angle  is 


M  =  0.75  Y 
Li 


(60) 


Equating  this  moment  to  the  moment  of  resistance  per  inch 


2-5x5x^1-5-13*13 


FIG.  15. 

of  length,  we  obtain  the  following  relation  between  the  bending 
stress  S"  and  M : 

Q//      18P 
8.-  -£ 


(61) 


L 

In  addition  to  this  flexural  stress  there  is  a  direct  stress  S't,  the 
magnitude  of  which  is 

Si  =  £  (62) 

The  summation  of  the  stresses  given  by  (61)  and  (62),  accord- 
ing to  the  conditions  of  the  problem  should  not  exceed  16,000; 
therefore 

P  =  IMOOL  ((J8) 

j.  v/ 


70  SPLICE  JOINT  [CHAP.  Ill 

Since  L  =  12  inches,  the  maximum  safe  load  that  the  angle  will 
stand  is,  according  to  (63),  equal  to  10,100  pounds. 

Comparing  this  load  with  those  determined  for  the  other 
methods  of  failure,  it  is  evident  that  the  10,100  pounds  is  the 
maximum  load  that  can  be  supported  safely  by  the  connection 
represented  in  Fig.  15. 

73.  Splice  Joint. — In  Fig.  16  is  shown  a  form  of  joint  used  in 
the  bottom  chord  of  a  Fink  roof  truss.  Four  members  are  joined 
together  by  means  of  a  vertical  gusset  plate  e  and  a  splice  plate  / 
underneath  the  outstanding  legs  of  the  bottom  chord  angles. 
Due  to  the  fact  that  a  Fink  truss  is  generally  shipped  in  four 


FIG.  16. 

pieces,  the  splice  joint  is  made  in  the  field.  In  the  joint  shown  in 
Fig.  16,  the  magnitude  of  the  loads  upon  the  members  a,  b,  c, 
and  d  are  30,100,  11,700,  13,000  and  17,700  pounds  respectively; 
it  is  required  to  design  the  complete  connection  assuming  the 
same  working  stresses  as  used  in  the  problem  of  Art.  72,  and 
furthermore,  that  no  plate  shall  have  a  thickness  less  than  J4  inch, 
(a)  Size  of  members. — In  Table  14  are  given  the  steps  that 
are  necessary  in  arriving  at  the  sizes  of  the  tension  members 
a,  c  and  d.  Attention  is  called  to  the  fact  that  the  sizes  of  the 
members  a  and  c  are  established  by  the  loads  given  in  Table  14 
and  not  by  those  given  above.  This  is  because  certain  mem- 
bers of  light  trusses  are  made  continuous.  According  to  certain 
specifications,  the  minimum  size  of  angles  used  is  2  by  2  by  J^ 
inch  while  according  to  others,  the  minimum  is  2}/£  by  2  by  J£ 
inch.  In  the  present  case  the  latter  size  is  adopted,  as  this  choice 
permits  the  use  of  %-inch  rivets  through  the  2^-inch  leg. 


ART.  73] 


SPLICE  JOINT 
TABLE  14. — TENSION  MEMBERS 


71 


Section  selected 

Truss 
mem- 

Max. load 

Allow, 
stress 

Required, 
area 

•vr_ 

Area 

ber 

Gross 

Net 

a 

36,600 

2.29 

2 

3^X2KXH 

2.88 

2.45 

c 

19,600 

16,000 

1.22 

2 

2^X2     XK 

2.14 

1.70 

d 

17,700 

1.11 

2 

2^X2     XX 

2.14 

1.70 

The  size  of  the  compression  member  b  is  arrived  at  in  a  general 
way  by  determining  the  allowable  unit  compressive  stress  by 
means  of  (25),  having  assumed  a  probable  cross-section  for  the 
member  in  question.  The  area  of  the  assumed  section  is  then 
compared  with  that  obtained  by  dividing  the  load  on  the  member 
by  the  calculated  unit  stress.  If  the  former  area  is  equal  to  or 
slightly  greater  than  the  latter,  the  section  assumed  is  safe.  In 
determining  the  area  of  a  compression  member  no  reduction  is 
made  for  the  rivet  hole,  as  it  is  assumed  that  the  rivet  in  filling 
up  the  hole  does  not  weaken  the  section. 

The  allowable  unit  compressive  stress  is  given  by  the  following 
expression  derived  directly  from  (25) : 

Se  =  16,000  -  70  -i  (64) 

in  which  I  denotes  the  length  of  the  member  in  inches  and  r  the 
least  radius  of  gyration  in  inches.  Generally  the  length  of  the 
compression  members  in  roof  trusses  should  not  exceed  125  times 
the  least  radius  of  gyration.  If,  as  in  a  roof -truss  problem,  it  is 
required  to  determine  the  size  of  a  series  of  compression  members, 
the  best  method  of  procedure  is  to  arrange  the  calculations  in 
tabular  form.  In  the  above  problem,  the  length  of  the  member 
b  is  93.8  inches,  and  the  thickness  of  all  plates  will  be  assumed  as 
Y±  inch. 

Assume  the  member  b  to  be  made  of  minimum  size  angles, 
namely,  two  2%  by  2  by  J^  inch  having  an  area  of  2.14  square 
inches.  The  least  radius  of  gyration  r  is  0.78  inch  when  the 
angles  are  arranged  back  to  back  with  a  14-inch  plate  between 
them.  This  gives  a  ratio  of  I  to  r  as  120  which  is  safe.  The 
allowable  working  stress  calculated  by  means  of  (64)  is  7600 

•     u    u  -     H,700 

pounds  per  square  inch;  hence,  the  required  area   is     7'   ~    or 

1.54  square  inches.     Since  the  area  of  the  members  chosen  is  in 


72  SPLICE  JOINT  [CHAP.  HI 

excess  of  the  calculated  area,  our  assumption  is  on  the  side  of 
safety. 

(b)  Number  of  rivets. — The  number  of  rivets  required  to 
fasten  each  of  the  members  b  and  c  to  the  gusset  plate  e  is  de- 
termined as  explained  in  Art.  70,  while  the  number  required  in 
the  members  a  and  d  depends  upon  various  assumptions  that  may 
be  made.  Among  these  are  the  following: 

1.  The  sum  of  the  horizontal  components  of  the  forces  in  the 
members  b  and  c,  which  is  equal  to  the  difference  between  the 
forces  acting  on  the  members  a  and  d,  is  transmitted  through 
the  gusset  plate  e  to  the  member  a;  hence,  the  number  of  rivets 
required  to  fasten  a  to  the  gusset  plate  is  based  on  this  force. 
It  follows  that  the  splice  plate  /  and  the  rivets  contained  therein 
must  be  designed  to  transmit  the  total  force  in  d.     The  vertical 
legs  of  the  member  d  must  also  be  riveted  to  the  plate,  but 
these  rivets  are  not  considered  as  a  part  of  the  splice. 

2.  Consider  that  all  of  the  rivets  in  the  connection  are  effective, 
that  is,  the  total  number  of  rivets  required  in  each  of  the  members 
a  and  d  must  be  based  on  the  load  transmitted  by  these  members. 
This  is  equivalent  to  making  the  gusset  plate  transmit  a  certain 
part,  say  approximately  one-half,  of  the  load  in  d,  and  the  re- 
mainder is  taken  up  by  the  splice  plate.     Due  to  the  fact  that  the 
splice  plate  is  riveted  to  a  and  d  by  an  even  number  of  rivets,  it 
frequently  happens  that  the  loads  taken  up  by  the  splice  and 
gusset  plates  are  far  from  being  equal.     The  method  of  pro- 
cedure is  shown  by  the  following  problem: 

The  size  of  the  members  will  permit  the  use  of  %-inch  rivets 
throughout,  except  in  the  splice  plate,  where  %-inch  rivets  must 
be  used.  We  shall  assume  that  four  %-inch  rivets  are  used  at 
each  end  of  the  splice  plate,  and  these  are  capable  of  transmitting 
4  X  2,045  or  8,180  pounds,  or  46  per  cent,  of  the  load  in  the 
member  d.  If  six  %-inch  rivets  are  used,  the  splice  plate  will 
then  transmit  69  per  cent,  of  the  load  in  d.  The  former  combina- 
tion is  the  one  selected,  as  by  its  use  the  entire  joint  can  be  made 
up  with  fewer  rivets  than  would  be  required  if  the  second  scheme 
were  used.  Now  the  remaining  load  in  d,  or  9,520  pounds,  is 
transmitted  through  the  gusset  plate.  The  load  in  the  member 
a  minus  the  load  transmitted  by  the  splice  plate  is  21,920  pounds; 
this  load  must  be  transmitted  through  the  gusset  plate  and  re- 

21  920 
quires   0  Lgrt   or  6  shop  rivets.     The  number  of  field  rivets  in 

'  9520 

the  vertical  legs  of  the  member  d  is  ^^  or  4. 


ART.  74] 


BOILER  BRACE 
11,700 


73 


The  member  b  requires  0  '    n  or  3  shop  rivets  while  the  member 

o,  /OU 

,    13,000 
c  needs  0  --n  or  4  shop  rivets. 

o,  /OU 

74.  Pin  Plates. — Not  infrequently  in  structural  work  forces 
are  transmitted  from  one  member  to  another  by  means  of  pins, 
and  in  such  cases  the  bearing  area  between  the  pin  and  members 
must  be  sufficient  to  transmit  the  load  safely.     A  common  case 
is  that  of  channels  through  the  webs  of  which  passes  a  pin.     In 
order  to  prevent  the  crushing  of  the  webs,  reinforcing  or  pin  plates 
must  be  riveted  to  them.     In  arriving  at  the  thickness  of  such 
pin  plates,  it  is  assumed  that  the  load  is  distributed  uniformly 
over  the  total  bearing  area,  and  that  each  plate  is  capable  of  tak- 
ing a  load  equal  to  the  total  load  multiplied  by  the  ratio  that  the 
thickness  of  the  plate  bears  to  the  total  thickness.     Knowing  the 
load  coming  upon  each  plate,  the  number  of  rivets -required  to 
fasten  it  to  the  web  of  the  channel  is  readily  obtained.     Another 
example  of  the  use  of  pin  plates  is  shown  in  the  reinforcing  of 
the  side  plates  of  crane  blocks. 

75.  Diagonal  Boiler  Brace. — In  Fig.  17  is  shown  a  form  of 
boiler  brace  used  for  connecting  the  unsupported  area  of  the  head 


FIG.  17. 

to  the  cylindrical  shell.  It  consists  of  a  round  rod  having 
flanged  or  flattened  ends  by  means  of  which  the  brace  is  riveted 
to  the  head  and  shell.  Due  to  the  action  of  the  steam  pressure, 
the  brace  may  fail  in  any  one  of  the  following  ways:  (1)  The  body 
of  the  brace  may  fail  by  tension ;  (2)  the  flanged  ends  at  the  head 
may  fail  due  to  flexure,  while  the  forged  end  at  the  shell  may  fail 
due  to  combined  bending  and  direct  tensile  stresses;  (3)  the  rivets 
may  fail  at  the  head  end ;  (4)  the  rivets  may  fail  at  the  shell  end. 


74  BOILER  BRACE  [CHAP.  Ill 

(a)  Failure  of  the  brace  body.  —  Letting  P  represent  the  force 
exerted  upon  the  brace  due  to  the  pressure  on  the  area  supported 
by  the  brace,  then  the  component  of  this  force  along  the  rod  is 
P  seca.  Hence  the  stress  in  the  rod  is  given  by  the  following 
expression  : 


(6)  Failure  of  the  brace  ends.  —  1.  Head  End.  —  The  end  at- 
tached to  the  boiler  head  may  fail  by  bending  of  the  outstanding 
legs.  If  2  e  represents  the  distance  between  the  two  rivets  as 
shown  in  the  figure,  then  the  stress  in  the  sections  adjacent  to 
the  rod  is 

S't  =  -(2e-a)  (66) 


As  usually  constructed  the  type  of  brace  shown  in  Fig.  17  is  con- 
siderably stronger  at  the  flanged  ends  than  in  the  body. 

2.  Shell  end.  —  At  the  shell  end  it  is  customary  to  investigate 
the  brace  merely  for  direct  tension.  Representing  the  width 
of  the  flanged  end  by  g  and  its  thickness  by  /,  then  the  tensile 
stress  is 

s"=  (67) 


(c)  Failure  of  the  rivets  at  the  head  end.  —  The  rivets  at  the 
head  end  of  the  brace  are  subjected  to  direct  tensile,  shearing, 
and  bending  stresses,  the  latter  two  of  which  are  generally  not 
considered  in  actual  calculations.  The  force  causing  the  tensile 
stress  in  the  rivets  is  the  total  force  P  minus  the  area  (I  X  c) 
multiplied  by  the  steam  pressure.  However,  since  the  shearing 
and  bending  stresses  are  not  considered,  it  is  customary  to  take 
the  total  force  P  as  coming  upon  the  two  rivets.  Hence  the 
tensile  stress  in  the  rivets  is 


(68) 
The  shearing  stress  coming  upon  the  rivets  is 

(69) 


If  it  is  desired  to  find  the  resultant  stress  due  to  the  combined 
effect  of  the  two  stresses  just  discussed,  use  the  equations  given 
in  Art.  17. 


ART.  75]  REFERENCES  75 

(d)  Failure  of  the  rivets  at  the  shell  end.  —  Due  to  the  pull  of 
the  brace,  the  rivets  at  the  shell  end  are  subjected  to  shearing, 
tensile,  and  bending  stresses.  The  first  of  these  stresses  is  gener- 
ally the  only  one  considered,  since  in  the  majority  of  cases  the 
direct  tensile  and  bending  stresses  are  small.  The  component  of 
the  force  in  the  rod  at  right  angles  to  the  rivets  has  a  magnitude 
of  P;  hence,  the  shearing  stress  in  the  rivets,  assuming  that  two 
rivets  are  used  to  fasten  the  brace  to  the  shell,  is 

9  P 

S.  =  (70) 


(e)  Allowable  stresses.  —  The  allowable  shearing  stresses  in  the 
rivets  vary  from  5,000  to  8,000  pounds  per  square  inch,  while  the 
permissible  tensile  stresses  in  the  diagonal  brace  proper  vary  from 
6,000  to  10,000  pounds  per  square  inch.  For  the  rivets  in  tension, 
the  allowable  stress  should  not  exceed  that  given  for  shearing. 

References 

Design  of  Steam  Boilers  and  Pressure  Vessels,  by  HAVEN  and  SWETT. 

Elements  of  Machine  Design,  by  KIMBALL  and  BARR. 

Elements  of  Machine  Design,  by  W.  C.  UNWIN. 

Die  Maschinen  Elemente,  by  C.  BACH. 

Mechanics  of  Materials,  by  M.  MERRIMAN. 

Steam  Boilers,  by  PEABODY  and  MILLER. 

Structural  Engineer's  Handbook,  by  M.  C.  KETCHUM. 


CHAPTER  IV 

FASTENINGS 
BOLTS,  NUTS,  AND  SCREWS 

76.  Forms  of  Threads. — The  threads  of  screws  are  made  in  a 
variety  of  forms  depending  upon  the  use  to  which  the  screws  are 
to  be  put.  In  general,  a  screw  intended  for  fastening  two  or 
more  pieces  together  is  fitted  with  a  thread  having  an  angular 
form,  while  one  intended  for  the  transmission  of  power  will  have 
the  threads  either  square  or  of  a  modified  angular  form. 

Two  common  forms  of  threads  used  for  screw  fastenings  are 
the  well-known  V  and  the  Sellers  or  United  States  Standard 
threads,  shown  in  Fig.  18(a)  and  (6),  respectively.  Both  forms 
are  strong  and  may  be  produced  very  cheaply.  Furthermore,  due 


to  their  low  efficiency,  they  are  well  adapted  for  screw  fastenings. 
The  proportions  of  these  threads  are  given  in  the  figures,  and,  as 
shown,  the  angle  used  is  60  degrees.  The  symbol  p  denotes  the 
pitch,  by  which  is  meant  the  axial  distance  from  a  point  on  one 
thread  to  the  corresponding  point  on  the  next  thread ;  or  in  other 
words,  the  pitch  is  the  distance  that  the  nut  advances  along  the 
axis  of  the  screw  for  each  revolution  of  the  nut.  Evidently,  the 
number  of  threads  per  inch  of  length  is  equal  to  the  reciprocal  of 
the  pitch  for  a  single-threaded  screw. 

76 


ART.  76]  FORMS  OF  THREADS  77 

(a)  Sellers  standard. — The  form  of  thread  shown  in  Fig.  18(6) 
is  recognized  as  the  standard  in  the  United  States,  though  the 
sharp  V  form  is  still  in  use.  Due  to  the  flattening  of  the  tops 
and  bottoms  of  the  V's  in  the  Sellers  standard,  this  form  is  much 
stronger  than  the  sharp  V  thread.  In  Table  15  are  given  the 
proportions  of  the  various  sizes  of  bolts  and  nuts  up  to  3  inches 
in  diameter,  based  on  the  Sellers  standard.  The  Sellers  system 
with  some  modifications  has  been  adopted  by  the  United  States 
Navy  Department.  Instead  of  using  different  proportions  for 
finished  and  unfinished  bolt  heads  and  nuts,  the  Navy  Depart- 
ment adopted  as  their  standard  those  given  for  rough  work,  thus 
permitting  the  same  wrench  to  be  used  for  both  classes  of  bolts. 
In  addition  to  this  change,  the  Navy  Department  has  adopted  a 
pitch  of  J4  mcn  f°r  aH  sizes  above  2%  inches,  which  does  not  agree 
with  the  Sellers  system. 

(6)  Standard  pipe  thread. — In  Fig.  18  (c)  is  shown  a  section  of  a 
standard  pipe  thread  which  may  also  be  considered  a  form  of  fas- 
tening, though  not  for  the  same  class  of  service  as  those  discussed 
above.  It  will  be  noticed  that  the  total  length  of  the  thread 
is  made  up  of  three  parts.  The  first  part  designated  as  A  in  Fig. 

18(c)  has  a  full  thread  over  a  tapered  length  of  -  ~>in 

which  D  represents  the  outside  diameter  of  the  pipe  and  n  the 
number  of  threads  per  inch.  The  second  part  B  has  two  threads 
that  are  full  at  the  root  but  imperfect  at  the  top  and  not  on  a 
taper.  The  part  C  includes  four  imperfect  threads.  The  total 
taper  of  the  threads  is  %  inch  per  foot,  or  the  taper  designated 
by  the  symbol  E  is  1  in  32.  It  should  be  remembered  that  gas 
pipe  goes  only  by  inside  measurement,  that  is,  by  the  nominal 
diameter.  The  actual  inside  diameter  varies  somewhat  from  the 
nominal,  but  only  the  latter  is  used  in  speaking  of  commercial 
pipe  sizes. 

(c)  Square  thread. — Three  forms  of  screw  threads  that  are 
well  adapted  to  the  transmission  of  power  are  shown  in  Fig.  19. 
The  square  thread  shown  in  Fig.  19 (a)  is  probably  the  most  com- 
mon, and  its  efficiency  is  considerably  higher  than  that  obtained 
by  the  use  of  V  threads.  It  has  serious  disadvantages  in  that  it 
is  very  difficult  to  take  up  any  wear  that  may  occur,  and  further- 
more, it  costs  considerably  more  to  manufacture.  The  pro- 
portions of  square  threads  have  never  been  standardized,  but  the 


78 


TABLE  OF  BOLTS  AND  NUTS  [CHAP.  IV 


to    to    to  to    to    to  to 

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a 
**S 

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to  to  to 

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XM   X<0   XOO   «X  OX  MX   \00   XM    \rH  t-X  V*  -HX  X'-   X5\   OX    v«    WX   OX   p4\ 

o 

£ 

"£  \M  VH  XM  XM      tO  tO  VH  10  tO 

Q  O  XW   O\   r-N   «\  \00   F-K  VH  V*  VH  XOO  MX  XrH  XOO   XfH  X^1   «5\  XOO   XW  XOO  ' 

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00 

Q  C^Tf<OO(NtOOO'-<'CO500»COi'-'I*'*O<N-HiN 

« 

ooooooooooo 
g  "        dddddddcJddd 

OOOO'4<CO<N'-HOO500 

H 

N-^l   S^N   \00  V*   \N   \rH  VflO   \-^  \PO  vflO    \^"  NOO  \N  \00    \-^  \00  N^*  \C«  \H" 

r-N  «\  eo\  t*\  <-N  w\  *o\  ccfs  t*\          wx  .-N  co\  ^S  «\  «\  r»\          ^^  »-N  eS\ 


ART.  76] 


ACME  THREAD 


79 


practice  of  Wm.  Sellers  and  Co.,  exhibited  in  Table  16,  may 
serve  as  a  guide. 

TABLE  16. — PROPORTIONS  OP  SELLERS  SQUARE  THREADS 


Size 

Threads 
per  inch 

Root 
diam. 

Size 

Threads 
per  inch 

Root 
diam. 

Size 

Threads 
per  inch 

Root 
diam. 

H 

10 

0.1625 

1M 

V/2 

1.0 

3 

1% 

2.5 

H 

8 

0.2658 

IK 

3 

1  .  2084 

3K 

1M 

2.75 

y* 

6^ 

0.3656 

IK 

2^ 

1.4 

3^ 

m 

2.962 

H 

5M 

0.466 

2 

2M 

1.612 

3H 

w 

3.168 

H 

5 

0.575 

2M 

2M 

1.862 

4 

iy2 

3.418 

H 

4^ 

0.6806 

2^ 

2 

2.0626 

1 

4 

0.7813 

2^ 

2 

2.3126 

(d)  Trapezoidal  thread. — The  trapezoidal  or  buttressed  thread 
shown  in  Fig.  19(6)  is  occasionally  used  for  the  transmission  of 
power  in  one  direction  only.  The  driving  face  of  the  thread  is  at 


right  angles  to  the  axis  of  the  screw,  while  the  back  face  makes  an 
angle  of  45  degrees,  as  shown  in  the  figure.  It  is  evident  that  the 
efficiency  of  this  form  of  thread  is  the  same  as  for  a  square  thread, 
while  its  strength  is  practically  that  of  the  V  thread.  No  stand- 
ard proportions  have  ever  been  devised  or  suggested,  except  those 
given  in  Fig.  19(6). 

(e)  Acme  thread. — The  Acme  thread  is  now  recognized  as  the 
standard  form  of  thread  for  lead  screws  and  similar  service,  since 
the  wear  can  readily  be  compensated  for  by  means  of  a  nut  split 


80 


MACHINE  BOLTS 


[CHAP.  IV 


lengthwise.  Its  efficiency  is  not  quite  as  high  as  that  of  a  square 
thread,  but  its  cost  of  production  is  less  since  dies  may  be  used  in 
its  manufacture.  The  form  of  the  standard  Acme  thread  is 
shown  in  Fig.  19(c),  and  in  Table  17  are  given  the  various  dimen- 
sions indicated  in  the  figure. 

TABLE  17. — PROPORTIONS  OF  ACME  STANDARD  THREADS 


Threads 
per 
inch 

a 

b 

t 

Threads 
per 
inch 

a 

b 

t 

10 

0.0319 

0.0371 

0.0600 

3 

0.1183 

0.1235 

0.1767 

9 

0.0361 

0.0413 

0.0655 

2^ 

0.1431 

0.1483 

0.2100 

8 

0.0411 

0.0463 

0.0725 

2 

0.1801 

0.1853 

0.2600 

7 

0.0478 

0.0529 

0.0814 

1H 

0.2419 

0.2471 

0.3433 

6 

0.0566 

0.0618 

0.0933 

IK 

0.2914 

0.2966 

0.4100 

5 

0.0689 

0.0741 

0.1100 

i 

0.3655 

0.3707 

0.5100 

4 

0.0875 

0.0927 

0.1350 

M 

0.7362 

0.7414 

1.0100 

SCREW  FASTENINGS 

In  general,  screw  fastenings  are  used  for  fastening  together 
either  permanently  or  otherwise  two  machine  parts.  To  ac- 
complish this  end,  the  following  important  forms  are  met  with  in 
machine  construction;  (a)  bolts;  (b)  cap  screws;  (c)  machine 
screws;  (d)  set  screws;  (e)  studs;  (f)  patch  bolts;  (g)  stay  bolts. 

77.  Bolts. — A  bolt  is  a  round  bar  one  end  of  which  is  fitted  with 
a  thread  and  nut,  while  the  other  end  is  upset  to  form  the  head. 
Bolts  are  well  adapted  for  fastening  machine  parts  rigidly;  but 
at  the  same  time  they  allow  the  parts  to  be  easily  disconnected. 
Whenever  conditions  or  surroundings  will  permit,  bolts  should 
be  used  for  fastening  machine  parts  together. 

(a)  Machine  bolts. — What  is  known  as  a  machine  bolt  has  a 
rough  body,  but  the  head  and  the  nut  may  be  rough  or  finished, 
as  desired.  Commercial  forms  of  machine  bolts  are  shown  in 
Fig.  20  (a)  and  (6).  The  heads  and  nuts  may  be  square  as  shown 
in  Fig.  20(a),  or  the  hexagonal  form  shown  in  Fig.  20(6)  may  be 
used.  The  standard  lengths  of  machine  bolts  as  given  in  the 
manufacturers'  catalogs  are  as  follows : 

1.  Between    1    and   5   inches   the   lengths  vary  by  J^-inch 
increments. 

2.  Between    5   and    12  inches,  the  lengths  vary  by  J 
increments. 


ART.  77] 


COUPLING  BOLTS 


81 


3.  Above  12  inches,  the  lengths  vary  by  1-inch  increments. 

Any  length  of  bolt,  however,  may  be  obtained,  but  odd  lengths 
cost  more  than  standard  lengths.'  The  length  of  the  threaded 
part  is  from  three  to  four  times  the  height  of  the  nut. 


(a) 


(c) 


TABLE   18. — COUPLING   BOLTS 


The  proportions  of  the  heads  and  nuts  as  used  on  standard 
machine  bolts  are  given  in  Table  15. 

(b)  Carriage  bolts. — The  carriage  bolt,  another  form  of  through 
bolt,  is  shown  in  Fig.  20 (c). 

Its  chief  use  is  in  connec- 
tion with  wood  construc- 
tion. 

(c)  Coupling      bolts. — A 
coupling    bolt  is  merely  a 
machine  bolt  that  has  been 
finished  all  over,  so  that  it 
may  be  fitted  into  reamed 
holes  of  the  same  diameter 
as  the  nominal  diameter  of 
the  bolt.     Coupling   bolts 
are    intended    for    use    in 

connection  with  the  best  forms  of  construction.  They  are  more 
expensive  to  produce  and  at  the  same  time  are  more  costly  to 
fit  into  place.  In  Table  18  are  given  the  dimensions  of  com- 
mercial sizes  of  such  bolts.  According  to  the  manufacturers' 


Head  and  nut 

Size 

Threads 
per 
inch 

Stock 
lengths 

Short 

Thick- 

diam. 

ness 

H 

13 

% 

H 

2     -4^ 

H 

11 

IHe 

% 

2     -5 

% 

10 

1M 

H 

2M-5M 

% 

9 

IKe 

y8 

2>i-5^ 

i 

8 

1% 

i 

2H-5H 

1M 

7 

1^6 

IX 

3     -6 

IK 

7 

2 

IK 

3^-6 

82 


AUTOMOBILE  BOLTS 


[CHAP.  IV 


lists,  coupling  bolts  are  made  with  hexagonal  heads  and  nuts 
only,  and  the  lengths  for  the  sizes  listed  in  Table  18  run  from 
2  inches  up  to  6  inches,  varying  by  quarter-inch  increments. 

(d)  Automobile  bolts. — The  various  bolts  discussed  in  the 
preceding  paragraphs  have  not  been  found  satisfactory  in  auto- 
mobile construction,  and  in  order  to  fulfill  the  requirements  of 
strength  and  space  limits  demanded  in  this  class  of  work,  the 
Society  of  Automobile  Engineers  has  adopted  a  special  standard. 
The  design  of  this  type  of  bolt  is  shown  in  Fig.  21,  and  the  data 


T3 

b 

«h 

-f  - 

r> 

li 

V 

I 

c— 

J 

r 

j^ 

-g 

^y 

—  e              e» 

j 

—  a  — » 


FIG.  21. 


in  Table  19  give  the  various  detail  dimensions  for  the  different 
sizes  that  have  so  far  been  standardized.  It  will  be  noticed  that 
the  heads  and  nuts  are  hexagonal,  and  that  the  thread  is  the 

TABLE  19. — S.  A.  E.  STANDARD  BOLTS  AND  NUTS 


Size 


Threads 
per 
inch 


Head  of  bolt 


Castellated  nut 


Cotter 
pin 


Plain 
nut 


H 

Ke 


28 
24 


KG 


K 


K 


KG 


H* 


KG 

y2 

KG 


24 
20 
20 
18 
18 
16 
16 
14 
14 


KG 
H 

H 

y8 

15/1 

i 

IKe 


»/32 


* 


KG 


27/64 
l5/32 
3%4 

KG 


KG 


KG 


X* 


IK 


12 
12 


12 
12 


KG 


2KG 


KG 


Sellers  standard.     Instead  of  using  the  same  pitch  as  that  recom- 


ART.  78] 


CAP  SCREWS 


83 


mended  in  the  Sellers  system,  a  finer  pitch  has  been  adopted; 
furthermore,  the  heads  and  nuts  are  made  somewhat  smaller. 
The  heads  are  slotted  for  a  screw  driver  and  the  nuts  are  recessed 
or  castellated  so  they  may  be  locked  to  the  bolt  by  means  of 
cotter  pins. 

78.  Screws. — Screws,  unlike  bolts,  do  not  require  a  nut,  but 
screw  directly  into  one  of  the  pieces  to  be  fastened,  either  the 
head  or  the  point  pressing  against  the  other  piece.  The  types  of 
screws  that  hold  the  pieces  together  by  the  pressure  exerted  by 


u 


(b) 


FIG.  22. 

the  head  of  the  screw  are  called  cap  screws  and  machine  screws, 
while  those  whose  points  press  against  a  piece  and  by  friction 
prevent  relative  motion  between  the  two  parts  are  called  set 
screws.  By  the  term  length  of  a  screw  is  always  meant  the 
length  under  the  head. 

(a)  Cap  screws. — Cap  screws  are  made  with  square,  hexagonal, 
round  or  filister,  flat  and  button  heads,  and  are  threaded  either 
United  States  Standard  or  with  V  threads.  The  various  forms 
for  heads  are  shown  in  Fig.  22,  and  in  Table  20  are  given  the 


84 


CAP  SCREWS 


[CHAP.  IV 


general  dimensions  of  the  commercial  sizes  that  are  usually  kept 
in  stock.  All  cap  screws,  except  those  with  filister  heads,  are 
threaded  three-fourths  of  the  length  for  one  inch  in  diameter  or 
less  and  for  lengtns  less  than  four  inches.  Beyond  these  dimen- 
sions, the  threads  are  cut  approximately  one-half  the  length. 
The  lengths  of  cap  screws  vary  by  quarter-inch  increments 
between  the  limits  given  in  Table  20. 

Cap  screws,  if  properly  fitted,  make  an  excellent  fastening  for 
machine  parts  that  do  not  require  frequent  removal.  To  insure 
a  good  fastening  by  means  of  cap  screws,  the  depth  of  the  tapped 
hole  should  never  be  made  less  than  one  and  one-half  times  the 
diameter  of  the  screw  that  goes  into  it.  In  cast  iron,  the  depth 
should  be  made  twice  the  diameter. 

TABLE  20. — STANDARD  CAP  SCREWS 


Square  head 

Hexagon  head 

Socket  head 

Threads 

Size 

per 

J 

inch 

Short 
diam. 

Thick- 
ness 

Length 

Short 
diam. 

Thick- 
ness 

Length 

Diam. 

Thick- 
ness 

Length 

K 

20 

H 

K 

} 

He 

K 

^ 

H 

K 

K-M 

He 

18 

He 

He 

H-8 

K 

He 

\  H-3 

He 

He 

%-3% 

H 

16 

K 

% 

J 

He 

% 

1 

He 

« 

H-4 

He 

14 

He 

He 

J^ 

He 

i  3/   4 

M 

He 

^*-4  K 

H 

13 

H 

K 

/  X4 

^i 

K 

/ 

H 

K 

H-6 

He 

12 

'Me 

He 

1     -4 

lMe 

He 

1     -4 

13/ie 

He 

I  1  _g 

H 

11 

H 

££ 

1     -4K 

% 

% 

1     -4J-2 

J^ 

% 

H 

i/ 

10 

H 

?/ 

.iK-4% 

1 

?/ 

1K-4H 

1 

H 

1K-6 

y* 
1 

8 

IK 

1 

1^-5 

IK 

1 

l%-5 

IK 

7 

196 

IK 

\2-5 

l^ 

IK 

>  2    -5 

IK 

7 

IK 

IK 

J 

IK 

IK 

Round  and  filister 

Button  head 

Flat  head 

Threads 

head 

Size 

per 

inch 

Diam. 

Thick- 
ness 

Length 

Diam. 

Thick- 
ness 

Length 

Diam. 

Thick- 
ness 

Length 

H 

40 

He 

K 

K-2H 

Ha 

H4 

^-1H 

K 

«-!« 

He 

24 

K 

He 

H-2H 

He 

^2 

^  -2 

2^ 

%-2 

K 

20 

J£ 

K 

%-3 

He 

J.<52 

2  -2K 

l%2 

H~2K 

He 

18 

He 

He 

H-3K 

He 

^2 

^  -2K 

H 

%-2% 

?£ 

16 

He 

% 

H-3M 

% 

He 

^  -2^ 

H 

%-3 

He 

14 

H 

He 

%-3H 

K 

^^ 

^  -3 

JHe 

1     -3 

K 

13 

H 

K 

^4-4 

xHe 

1^2 

1     -3 

H 

1K-3 

He 

12 

1He 

He 

i    -4K 

JHe 

1^2 

1K-3 

1 

1K-3 

« 

11 

% 

££ 

1K-4M 

1 

^ 

1M-3 

IK 

1^-3 

% 

10 

1 

K 

1K-4H 

IK 

% 

l^i-3 

1% 

2     -3 

H 

9 

IK 

?•£ 

1^-5 

1 

8 

IK 

1 

2     -5 

ART.  78] 


MACHINE  SCREWS 


85 


(b)  Machine   screws. — Machine  screws  are  strictly  speaking 
cap  screws,  but  the  term  as  commonly  used  includes  various  forms 
of  small  screws  that  are  provided  with  a  slotted  head  for  a  screw- 
driver.    The  sizes  are  designated  by  gauge  numbers  instead  of 
by  the  diameter  of   the  body.     The  usual   forms  of   machine 
screws  are  shown  in  Fig.  23, 
and    in    Table    21   are  given 
the  dimensions  of  stock  sizes. 

There     are     no     accepted 
standards,      each     manufac-    ^qip^ 

1.1,1 


turer  having  his  own.  It 
should  also  be  observed  that 
machine  screws  have  no 
standard  number  of  threads, 
hence  in  dimensioning  these 
screws,  always  give  the  num- 
ber of  the  screw,  the  number 
of  threads  and  the  length, 

thus  No.  30  —  16  X  1M  inches  M.  Sc.  It  may  be  noted  that 
machine  screws  larger  than  No.  16  are  not  used  extensively  in 
machine  construction;  for  larger  diameters  than  No.  16,  use 
cap  screws. 


(a) 


(b) 

FIG.  23. 


(C) 


(a) 


(b) 


(c)  (d) 

FIG.  24. 


(f) 


The  American  Society  of  Mechanical  Engineers  has  adopted 
a  uniform  system  of  standard  dimensions  for  machine  screws, 
but  as  yet  they  are  not  in  universal  use  in  this  country.  The 
report  of  the  committee  which  was  appointed  to  draw  up  such 
standards  may  be  found  on  page  99  of  volume  29  of  the 
Transactions. 

(c)  Set  screws. — Set  screws  are  made  with  square  heads  or 
with  no  heads  at  all",  and  may  be  obtained  with  either  United 
States  Standard  or  V  threads.  The  short  diameter  of  the  square 


86 


TABLE  OF  MACHINE  SCREWS  [CHAP.  IV 


^Hi-l(N<N<N01?OCOC«3Tf<Tj<iOkOCOOt-COt^t^ 
OOOOOOOOOOOOOOOOOOO 

ddodo'ddodddddddodoo 


epuaij 


SB  amiss  aqj, 


OOOOOOOo222!2222<N2c^c3 

odddddddddddddddddd 


OOOOOOOOOOOOOOOOOOO 


ooooooooooooooooooo 


sp^aq  aa^sijg  uo  SB  auiBS  aqj, 


COt-OOOOCiOi-H'-i 

OOOOOT-lrH,-! 


ooooooooooooooooooo 


OOOOOOOOOOOOOOOOOOO 


COCO'*^'OCDCOt>t>'OOO5 


ooooooooooooooooooo 


CCCOCCfOC»3'*Tf<'<tiTf*iO»OCOOb-t>.t^OOGOO> 

ooooooooooooooooooo 
ddddddddddddddddddd 


ooooooooooooooooooo 


O 


,_(,-H,_i,_i<N<N<N<N<NOOeOTfiTf<'<t|iOiOCOCOCO 
OOOOOOOOOOOOOOOOOOO 

ddddoddo'ooooo'dbo'ooo 


OOOOOOOOOOOOOOOOOOO 


Threads 
per  inch 


00  COOOW  Oo2 


ooooooooooooooooooo 


0M  2J"»  00  0  N  ^  g  00  0 


ART.  78] 


SET  SCREWS 


87 


heads  as  well  as  the  height  of  the  heads  is  made  equal  to  the 
diameter  of  the  body  of  the  screw.  The  commercial  lengths  of 
set  screws  having  heads  vary  from  %  to  5  inches  by  quarter  inches. 
The  headless  set  screws  shown  in  Fig.  24  (d)  to  (/)  are  made 
only  in  the  following  sizes:  %  by  %  inch;  ^  by  %6  inch;  %  by 
iy\_6  inch;  and  %  by  %  inch. 

The  principal  distinguishing  feature  of  set  screws  is  the  form  of 
the  point.  The  points  are  generally  hardened.  Only  cup  and 
round  point  set  screws  (see  Fig.  24  (a)  and  (6))  are  regular,  all 
other  types  being  considered  special.  Set  screws  used  as  fasten- 
ings are  not  entirely  satisfactory  for  heavy  loads,  and  hence  should 
only  be  used  on  the  lighter  loads.  The  cup  point  shown  in  Fig. 
24 (a)  has  a  disadvantage  in  that  it  raises  a  burr  on  the  shaft  thus 
making  the  removal  of  the  piece,  such  as  a  pulley,  more  difficult. 
In  place  of  the  cup  point,  the  conical  point  shown  in  Fig.  24 (e) 
is  frequently  used,  but  this  necessitates  drilling  a  conical  hole  in 
the  shaft,  which  later  on  may  interfere  with  making  certain  de- 
sirable adjustments. 

To  obtain  the  appropriate  size  of  set  screw  for  a  given  diameter 
of  shaft,  the  following  empirical  formula  based  upon  actual  in- 
stallations may  be  found  useful: 


diameter  of  set  screw  =  -  +  ^{Q  inch, 


(71) 


TABLE    22. — SAFE    HOLDING   CA- 
PACITIES OP  SET  SCREWS 


in  which  d  represents  the  diameter  of  the  shaft. 

The  question  of  the  holding  capacity  of  set  screws  has  received 
little  attention  and  about  the  only 
information  available  is  that  pub- 
lished by  Mr.  B.  H.  D.  Pinkney 
in  the  American  Machinist  of  Oct. 
15,  1914.  His  results  are  based 
upon  some  experiments  with  y±- 
and  J^-inch  set  screws,  in  which 
he  found  that  the  latter  size  had  a 
capacity  of  five  times  the  former. 
With  this  fact  as  a  basis,  Mr. 
Pinkney  calculated  the  data  given 
in  Table  22.  Experience  with  the  headless  variety  of  set  screws 
seems  to  indicate  that  due  to  the  difficulty  of  screwing  up,  the 
holding  power  is  somewhat  less  than  for  the  cup  and  flat  point 
type. 


Size 

Capacity, 
pounds 

Size 

Capacity 
pounds 

H 

100 

H 

840 

He 

168 

H 

1,280 

% 

256 

H 

1,830 

K6 

366 

i 

2,500 

H 

500 

IK 

3,288 

KG 

658 

IK 

4,198 

88 


PATCH  BOLTS 


[CHAP.  IV 


(d)  Studs. — A  stud  is  a  bolt  in  which  the  head  is  replaced  by 
a  threaded  end,  as  shown  in  Fig.  25.     It  passes  through  one  of 
the  parts  to  be  connected,  and  is  screwed  into  the  other  part, 
thus  remaining  always  in  position  when  the  parts  are  disconnected. 
With  this  construction  the  wear  and  crumbling  of  the  threads 
in  a  weak  material,  such  as  cast  iron,  are  avoided.     Studs  are 

usually  employed  to  secure 
the  heads  Jof  cylinders  in  en- 
gines and  pumps. 

There  is  no  standard  for 
the  length  of  the  threaded 
ends  of  studs;  hence,  the 
length  must  always  be  speci- 
fied. Studs  may  be  obtained  finished  at  B  or  rough,  and  the 
ends  threaded  either  with  United  States  Standard  or  V  threads. 
The  commercial  lengths  carried  in  stock  vary  from  \Y±  to  6 
inches  by  quarter  inches  for  the  finished  studs.  For  the  rough 
studs,  the  lengths  vary  from  Y^  to  4  inches  by  quarter  inches,  and 
from  4  to  6  inches  by  half  inches.  Usually  one  end  is  made  a 
tight  fit,  while  the  other  is  of  standard  size. 

(e)  Patch  bolts. — A  form  of  screw  commonly  called  a  patch 
bolt  is  shown  in  Fig.   26 (a);  its  function  is  that  of  fastening 
patches  on  the  sheets  of  boilers.     The  application  of  a  patch 


FIG.  25. 


(b) 
FIG.  26. 

bolt  is  illustrated  in  Fig.  26  (fr).  Patch  bolts  should  be  used  only 
when,  due  to  the  location  of  the  patch,  it  is  impossible  to  use 
rivets,  as  for  example  on  the  water  leg  of  a  locomotive  boiler. 
As  shown  in  Fig.  26(6),  patch  bolts  are  introduced  from  the  side 
exposed  to  the  fire  and  are  screwed  home  securely.  The  head, 
by  means  of  which  they  are  screwed  up,  is  generally  twisted  off 
in  making  the  fastening.  Instead  of  having  the  form  and  num- 
ber of  threads  according  to  the  United  States  Standard,  all  stock 
sizes  have  12  threads  per  inch  of  the  sharp  V  type. 


ART.  79] 


STAY  BOLTS 


89 


79.  Stay  Bolts. — (a)  Stay  bolts  are  fastenings  used  chiefly  in 
boiler  construction.  Due  to  the  unequal  expansion  and  con- 
traction of  the  two  plates  that  are  connected,  stay  bolts  are 
subjected  to  a  peculiar  bending  action  in  addition  to  a  direct 
tension.  As  a  result  of  the  relative  motion  between  the  two 
connected  plates,  stay  bolts  develop  small  cracks  near  the  inner 
edge  of  the  sheets.  These  cracks  eventually  cause  complete 
rupture,  though  it  may  not  be  noticed  until  the  plates  begin  to 
bulge.  Three  types  of  stay  bolts  are  shown  in  Fig.  27,  the  first 


(a) 


(b) 


Fia.  27. 

of  which  is  used  extensively  on  small  vertical  and  locomotive 
types  of  boilers.  To  provide  some  slight  degree  of  flexibility 
and  thereby  decrease  the  danger  of  cracking  near  the  plates, 
stay  bolts  are  made  as  shown  in  Fig.  27(6). 

According  to  the  Code  of  Practical  Rules,  covering  the  con- 
struction and  maintenance  of  stationary  boilers,  recently  adopted 
by  the  American  Society  of  Mechanical  Engineers,  "each  end  of 
stay  bolts  must  be  drilled  with  a  J^g-inch  hole  to  a  depth  extend- 
ing 3^2  inch  beyond  the  inside  of  the  plates,  except  on  small  vertical 
or  locomotive-type  boilers  where  the  drilling  of  the  stay  bolts 
shall  be  optional."  The  object  of  these  holes  is  to  give  some 
indication  of  a  rupture  by  the  leakage  of  the  fluid. 

In  Fig.  27  (c)  is  shown  one  of  the  various  types  of  so-called 
flexible  stay  bolts  used  in  locomotive  boilers. 


90  NUT  LOCKS  [CHAP.  IV 

(6)  Stresses  in  stay  bolts.  —  The  area  of  the  surface  supported 
by  a  stay  bolt  depends  principally  upon  the  thickness  of  the 
plates  and  the  fluid  pressure  upon  the  surface.  Quoting  from  the 
Code  of  Rules  adopted  by  the  American  Society  of  Mechanical 
Engineers,  "the  pitch  allowed  for  stay  bolts  on  a  flat  surface  and 
on  the  furnace  sheets  of  an  internally  fired  boiler  in  which  the 
external  diameter  of  the  furnace  is  over  38  inches,  except  a  corru- 
gated furnace,  or  a  furnace  strengthened  by  an  Adamson  ring  or 
equivalent,"  may  be  determined  by  the  following  formula,  but 
in  no  case  should  it  exceed  8%  inches  : 


t,  (72) 

in  which 

C  =  constant  having  a  value  of  66. 

P  =  working  pressure  in  pounds  per  square  inch. 

t  =  thickness  of  plate  in  sixteenths  of  an  inch. 

In  addition  to  the  formula  just  given,  the  above-mentioned 
Code  of  Rules  contains  tables  and  other  formulas  pertaining  to 
the  subject  of  staying  surfaces  that  may  be  found  useful  in  de- 
signing pressure  vessels. 

Having  determined  the  pitch  of  the  stay  bolts,  a  simple  cal- 
culation will  give  the  magnitude  of  the  load  coming  upon  each 
bolt.  Dividing  this  load  by  the  allowable  stress,  the  result  is 
the  area  at  the  root  of  the  thread.  For  mild-steel  or  wrought- 
iron  stay  bolts  up  to  and  including  \Y±  inches  in  diameter,  the 
American  Society  of  Mechanical  Engineers  recommends  that  the 
allowable  stress  shall  not  exceed  6,500  pounds  per  square  inch, 
and  for  larger  diameters  7,000  pounds  per  square  inch  is  recom- 
mended. The  majority  of  screwed  stay  bolts  have  12  threads 
per  inch  of  the  V  type,  though  the  United  States  Standard  form 
is  also  used. 

80.  Nut  Locks.  —  Since  nuts  must  have  a  small  clearance 
in  order  to  allow  them  to  turn  freely,  they  have  a  tendency  to  un- 
screw. This  tendency  is  especially  evident  in  the  case  of  nuts 
subjected  to  vibration.  In  order  to  prevent  unscrewing,  a  great 
many  different  devices  have  been  originated,  a  few  of  which  are 
shown  in  Figs.  28  to  30  inclusive. 

(a)  Lock  nut.  —  The  cheapest  and  most  common  locking  de- 
vice is  the  lock  nut  shown  in  Fig.  28.  Two  nuts  are  used,  but 
it  is  not  necessary  that  both  of  these  shall  be  of  standard  thick- 


ART.  80] 


LOCK  NUT 


91 


ness,  as  frequently  the  lower  nut  is  made  only  one-half  as  thick 
as  the  upper  one.  Some  engineers  maintain  that  the  lower  nut 
should  be  standard  thickness  while  the  upper  one  could  be  thinner. 
The  following  analysis,  due  to  Weisbach,  shows  that  conditions 
might  arise  for  which  the  first  arrangement  would  answer,  while 
for  other  conditions,  the  second  arrangement  would  be  the  proper 
one  to  use. 

We  shall  assume  that  the  lower  nut  in  Fig.  28 (a)  has  been 
screwed  down  tight  against  the  cap  c  of  some  bearing.  Denote 
the  pressure  created  between  the  nut  b  and  the  cap  c  by  the 
symbol  P.  Now  screw  down  the  upper  nut  a  against  b  as  tightly 
as  the  size  of  the  stud  or  bolt  d  will  permit,  thus  developing  a 
pressure  between  the  two  nuts,  which  at  the  same  time  produces 
a  tensile  stress  in  that  part  of  the  stud  d  that  comes  within  the 
limits  of  action  of  the  two  nuts.  Designate  the  magnitude  of 
this  pressure  between  the  nuts  by  the  symbol  PQ.  Considering  the 


FIG.  28. 

forces  acting  upon  the  nut  fc,  it  is  evident  that  the  force  PQ  acts 
downward,  while  the  force  P  acts  upward,  and  the  resultant 
force  having  a  magnitude  P  —  PQ  acts  upon  the  threads  of  the 
stud.  Now  the  direction  of  this  resultant  depends  upon  the 
magnitudes  of  P  and  PQ.  If  P>PQ  the  resultant  force  on  the 
threads  of  the  stud  is  upward,  or  in  other  words  the  upper 
surfaces  of  the  threads  in  the  nut  b  come  into  contact  with  the 
lower  surfaces  of  the  threads  on  the  stud.  From  this  it  follows 
that  when  P>P0,  the  nut  b  should  be  of  standard  thickness  as 
shown  in  Fig.  28(a),  since  it  alone  must  support  the  axial  load. 
Let  us  consider  the  case  when  P0>P;  we  shall  find  that  the 
resultant  force  on  the  threads  is  downward,  thus  indicating  that 


92 


COLLAR  NUT 


[CHAP.  IV 


the  lower  surface  of  the  threads  in  the  nut  b  bear  on  the  upper 
surfaces  of  the  threads  on  the  stud;  hence,  the  upper  nut  must 
take  the  axial  load  and  for  that  reason  should  be  made  of  standard 
thickness  as  shown  in  Fig.  28(&). 

Now  consider  another  case  might  arise,  namely  in  which 
PO  =  P.  It  is  evident  that  the  resultant  is  zero,  thereby  showing 
that  no  pressure  exists  on  either  the  upper  or  lower  surfaces  of 
the  thread ;  hence,  the  nut  a  carries  the  axial  load  P. 

On  the  spindles  of  heavy  milling  machines  and  other  machine 
tools,  the  double  lock  nut  is  used  to  a  great  extent.  The  nuts  are 
made  circular  rather  than  hexagonal  and  are  fitted  with  radial 
slots  or  holes  for  the  use  of  spanner  or  pin  wrenches. 

(b)  Collar  nut. — The  collar  nut,  shown  in  Fig.  29 (a),  has  been 
used  very  successfully  in  heavy  work.  The  lower  part  of  the  nut 


FIG.  29. 


is  turned  cylindrical,  and  upon  the  surface  a  groove  is  cut.  The 
cylindrical  part  of  the  nut  fits  into  a  collar  or  recess  in  the  part 
connected.  This  collar  is  prevented  from  turning  by  a  dowel 
pin  as  shown  in  the  figure.  A  set  screw  fitted  into  the  collar 
prevents  relative  motion  between  the  latter  and  the  nut.  In 
connecting  rods  of  engines,  for  example,  where  the  bolt  comes 
near  the  edge  of  the  rod,  the  bolt  hole  is  counterbored  to  receive 
the  cylindrical  part  of  the  nut,  and  the  set  screw  for  locking  the 
nut  is  fitted  directly  into  the  head  of  the  rod. 

The  following  formulas  have  proved  satisfactory  in  propor- 
tioning collar  nuts  similar  to  that  shown  in  Fig.  29 (a): 


ART.  80] 


SPLIT  NUT 


93 


A  =  2.25  d  +  %6  inch 

B  =  1.5  d 

C  =  1.45  d 

D  =  0.75  d 

E  =  0.55  d 

F  =  0.2  d  +  Me  inch 

£  =  0.1  d  +  0.1  inch 


(73) 


(c)  Castellated  nut. — Another  effective  way  of  locking  nuts, 
used  extensively  in  automobile  construction,  is  shown  in  Fig. 
21.  It  is  known  as  the  castellated  nut,  and  the  commercial  sizes 
correspond  to  the  sizes  of  automobile  bolts  discussed  in  Art. 
77 (d).  Attention  is  directed  to  the  fact  that,  due  to  the  necessity 
of  turning  the  nut  through  60  degrees  between  successive  lock- 
ing positions,  it  may  be  impossible  to  obtain  a  tight  and  rigid 


FIG.  30. 

connection  without  inducing  a  high  initial  stress  in  the  bolt. 
The  general  proportions  of  the  standard  castellated  nuts  approved 
and  recommended  by  the  Society  of  Automobile  Engineers  are 
given  in  Table  19. 

(d)  Split    nut. — The    double  nut  method   of  locking  is  not 
always  found  convenient  due  to  restricted  space,  and  in  such 
places,  the  forms  of  nut  locks  shown  in  Fig.  30  have  been  found 
very  satisfactory.     In  Fig.  30(6)  is  illustrated  a  hexagonal  nut 
having  a  saw  cut  extending  almost  to  the  center.     By  means  of  a 
small  flat-head  machine  screw  fitted  into  one  side  of  the  nut,  the 
slot  may  be  closed  in  sufficiently  to  clamp  the  sides  of  the  thread. 
The  nut,  instead  of  being  hexagonal  in  form,  may  be  made  cir- 
cular, and  should  then  be  fitted  with  radial  slots  or  holes  for  a 
spanner  wrench. 

(e)  Spring  wire  lock. — The  spring  wire  lock  shown  in  Fig.  30  (a) 
is  another  locking  device  adapted  to  a  restricted  space.     This  is 
a  very  popular  nut  lock  for  use  with  the  various  types  of  ball 


94 


WASHERS 


[CHAP.  IV 


TABLE  23. — PLAIN  LOCK  WASHERS 


bearings.  The  spring  wire  requires  the  drilling  of  a  hole  in  the 
shaft,  and  in  case  any  further  adjustment  is  made  after  the  nut 
is  fitted  in  place,  it  requires  drilling  a  new  hole.  A  series  of 
such  holes  will  weaken  the  shaft  materially. 

(/)  Lock  washer. — A  nut  lock  used  considerably  on  railway 
track  work,  and  within  recent  years  in  automobile  work,  is  shown 
in  Fig.  29(6).  It  consists  essentially  of  one  complete  turn  of  a 
helical  spring  placed  between  the  nut  and  the  piece  to  be  fastened. 
When  the  nut  is  screwed  down  tightly,  the  washer  is  flattened  out 
and  its  elasticity  produces  a  pressure  upon  the  nut,  thereby  pre- 
venting backing  off.  In  Table  23  is  given  general  information 

pertaining  to  the  standard  light 
and  heavy  lock  washers  adopted 
by  the  Society  of  Automobile 
Engineers. 

81.  Washers. — The  function  of 
a  washer  is  to  provide  a  suitable 
bearing  for  a  nut  or  bolt  head. 
Washers  should  not  be  used  unless 
the  hole  through  which  the  bolt 
passes  is  very  much  oversize,  or 
the  nature  of  the  material  against 
which  the  nut  or  bolt  head  bears 
necessitates  their  use.  For  com- 
mon usage  with  machine  parts, 
wrought-iron  or  steel-cut  washers 
are  the  best.  When  the  material 
against  which  the  nut  bears  is  rel- 
atively soft,  such  as  wood  for  ex- 
ample, the  bearing  pressure  due  to  the  load  carried  by  the  bolt 
should  be  distributed  over  a  considerable  area.  This  is  accom- 
plished by  the  use  of  large  steel  or  cast-iron  washers. 

Washers  are  specified  by  the  so-called  nominal  diameter,  by 
which  is  meant  the  diameter  of  the  bolt  with  which  the  washer 
is  to  be  used. 

82.  Efficiency  of  V  Threads. — Before  discussing  the  stresses 
induced  in  bolts  and  screws  due  to  the  external  loads  and  to 
screwing  up,  it  is  necessary  to  establish  an  expression  for  the 
probable  efficiency  of  screws. 


Size  of 
bolt 

Section  of  washer 

Light  service 

Heavy  service 

Width 

Thick- 
ness 

Width 

Thick- 
ness 

KG 

KG 

H* 

y± 

H 

KG 

KG 

H 

KG 

H 

%2 

KG 

5/32 

H2 

KG 

%2 

KG 

5/32 

%2 

y* 

5/32 

KG 

5/32 

3/32 

KG 

KG 

H* 

% 

KG 

3/B2 

K 

KG 

K 

KG 

%2 

H 

M 

14 

H 

H 

y± 

H 

i 

H 

H 

X 

H 

ART.  82] 


EFFICIENCY  OF  V  THREADS 


95 


Let  N  =  unit  normal  pressure. 

Q  =  axial  thrust  upon  the  screw. 

d    =  mean  diameter  of  the  screw. 

p    =  pitch  of  the  thread. 

a    =  angle  of  rise  of  the  mean  helix. 

j8    =  angle  that  the  side  of  the  thread  makes  with  the 

axis  of  the  screw. 

/*'  =  coefficient  of  friction  between  the  nut  and  screw. 
77    =  efficiency. 

Consider  a  part  of  a  V-threaded  screw,  as  shown  in  Fig.  31,  in 
which  the  section  CDE  is  taken  at  right  angles  to  the  mean  helix 
AO.  The  line  OF  represents  the  line  of  action  of  the  normal 


FIG.  31. 

pressure  N  acting  upon  the  thread  at  the  point  0,  and  OY  is 
drawn  parallel  to  the  axis  of  the  screw. 

The  vertical  component  of  the  normal  pressure  N  acts  down- 
ward and  has  a  magnitude  of  N  cos  7.  -The  vertical  component 
of  the  force  of  friction  due  to  the  normal  pressure  N  acts  upward, 
and  its  magnitude  is  //'JV  sin  a.  The  algebraic  sum  of  these  two 
vertical  forces  gives  the  magnitude  of  the  component  of  Q  acting 
at  the  point  0.  Thus 

Q  =  2>N  (cos  7  —  //sin  a), 
from  which  the  total  normal  component  is 

Q 


2N  = 


(74) 


cos  7  —  n  sin  a 
In  one  revolution  of  the  screw,  the  applied  effort  must  be  capa- 


96  EFFICIENCY  OF  V  THREADS  [CHAP.  IV 

ble  of  doing  the  useful  work  Qp  and  overcoming  the  work  of 
friction.  Denoting  the  total  work  put  in  by  the  effort  in  one 
revolution  of  the  screw  by  the  symbol  Wt,  we  find  that 


cos  7  —  /*  sin  a 
Substituting  the  value  of  p  =  ird  tan  a  in  (75),  we  get 

Wt  =  irdQ  [tan  a  +  -  M/SGC  %    -1  (76) 

cos  7  —  ju  sm  aJ 

Now  to  determine   a  relation  between  the  angles  a,  /3  and  7, 
we  make  use  of  a  theorem  in  Solid  Analytical  Geometry,  namely, 

cos2  7  +  cos2/?  +  cos2||  —  a  1   =  1, 

from  which 

cos  7  =  V/cos2a  -  cos2/3  (77) 

Substituting  (77)  in  (76),  the  following  expression  for  the 
total  work  required  per  revolution  of  the  screw,  in  order  to  raise 
the  load  Q,  is  obtained: 


Wt  =  irdQ  [tan  a  +  -=  M/SGC  * 

V  cos2  a  —  cos2j3  —  //sin  a  \ 


(78) 


By  definition,  the  efficiency  is  the  ratio  of  the  useful  work  to 
the  total  work;  hence,  for  the  F-threaded  screw 

tan  a  , 

77  =  -  —  -f  —  (79) 

//'sec  a 
tan  a  +       .— 

V  cos2  a  —   cos2  18  —  //sin  a 

Very  often  it  is  desirable  to  determine  the  magnitude  of  the 
effort  P  required  at  the  end  of  a  lever  or  wrench.  Representing 
the  length  of  the  lever  by  L,  and  equating  the  work  done  by 
P  in  one  revolution  to  the  total  work  done,  we  find  that 

(80) 


STRESSES  IN  SCREW  FASTENINGS 

To  arrive  at  the  proper  dimensions  of  bolts,  screws  and  studs 
used  as  fastenings,  it  is  important  to  consider  carefully  the  follow- 
ing stresses: 


ART.  83]  STRESSES  IN  SCREWS  97 

(a)  Initial  stresses  due  to  screwing  up. 
(6)  Stresses  due  to  the  external  forces. 
(c)  Stresses  due  to  combined  loads. 

83.  Stresses  Due  to  Screwing  Up.  —  The  stresses  induced  in 
bolts,  screws  and  studs  by  screwing  them  up  tightly  are  a  tensile 
stress  due  to  the  elongation  of  the  bolt,  and  a  torsional  stress  due 
to  the  frictional  resistance  on  the  thread.     To  determine  the  mag- 
nitude of  the  resultant  stress  induced  in  a  fastening  subjected  to 
these  stresses,  combine  them  according  to  Art.  17.     For  screws  less 
than  %  inch  in  diameter,  the  stresses  induced  by  screwing  up 
depend  so  much  upon  the  judgment  of  the  mechanic  that  it  is 
useless  to  attempt  to  calculate  their  magnitude. 

Experiments  on  screws  and  bolts  have  been  made  with  the 
hope  that  the  results  obtained  would  furnish  the  designer  some 
idea  as  to  the  magnitude  of  the  stresses  due  to  screwing  up.  As 
might  be  expected,  the  results  varied  within  rather  wide  limits 
so  that  no  specific  conclusions  could  be  drawn;  however,  all 
such  tests  seemed  to  show  that  the  stresses  are  high,  generally 
higher  than  those  due  to  the  external  forces  and  very  frequently 
running  up  to  about  one-half  of  the  ultimate  strength  of  the  bolt. 

84.  Stresses  Due  to  the  External  Forces.  —  (a)  Direct  stress.  — 
Bolts,  screws,  and  studs,  as  commonly  used  for  fastening  machine 
parts,  are  subjected  to  a  direct  tensile  stress  by  the  external 
forces  coming  upon  them;  but  occasionally  the  parts  fastened 
will  produce  a  shearing  action  upon  the  fastening. 

Assuming  that  a  certain  force  Q  causes  a  direct  tensile  stress 
in  a  bolt  or  screw,  it  is  evident  that  the  weakest  section,  namely 
that  at  the  root  of  the  thread,  must  be  made  of  such  a  diameter 
that  the  stress  induced  will  not  exceed  the  allowable  tensile 
stress.  Calling  the  diameter  at  the  root  of  the  thread  do,  we 
obtain  from  (3) 


Table  15  gives  the  values  of  d0  for  the  various  sizes  of  the 
Sellers  standard  threads.  Since  this  table  also  gives  the  area  at 
the  root  of  the  thread,  the  calculations  for  the  size  of  a  bolt  for  a 
given  load  Q  is  considerably  simplified  by  finding  the  ratio  of  Q 
to  St  which  is  really  the  root  area  of  the  required  size  of  bolt; 
then  select  from  Table  15  the  diameter  corresponding  to  the 
area. 


98 


STRESSES  IN  SCREWS 


[CHAP.  IV 


Screws  subjected  to  a  shearing  stress  should  be  avoided  as  far 
as  possible.  However,  such  an  arrangement  can  be  used  success- 
fully by  the  use  of  dowel  pins  fitted  accurately  into  place  after 
the  screws  have  been  fitted.  There  are  many  places  where 
dowel  pins  cannot  be  used,  and  for  such  cases  it  is  suggested  that 
the  body  of  the  bolt  or  screw  be  made  an  accurate  fit  in  the 
holes  of  the  parts  to  be  fastened. 

Assuming  as  above  that  the  external  force  coming  upon  the 
bolt  is  Q,  and  that  the  allowable  shearing  stress  is  S8,  then  it 
follows  that 

fT~Fi 

(82) 

(b)  Tension  due  to  suddenly  applied  loads. — The  loads  pro- 
ducing the  stresses  discussed  in  the  preceding  paragraphs  were 
considered  as  steady  loads;  however,  bolts  and  screws  are  used 
in  many  places  where  the  loads  coming  upon  them  are  in  the 


(b) 


FIG.  32. 


nature  of  shocks,  as  for  example  in  the  piston  rod  of  a  steam 
hammer,  and  in  the  bolts  of  engine  connecting  rods.  Such  bolts 
must  then  be  designed  so  as  to  be  capable  of  resisting  the  shocks 
due  to  the  suddenly  applied  loads  without  taking  a  permanent 
set.  Now  since  the  energy  of  the  suddenly  applied  load  must  be 
absorbed  by  the  bolt,  and  as  the  measure  of  this  energy  is  the 
product  of  the  stress  induced  and  the  total  elongation,  it  is  evi- 
dent that  the  stress  may  be  reduced  by  increasing  the  elongation. 
Increasing  the  elongation  may  be  accomplished  in  several  ways, 
among  which  are  the  following: 

1.  Turn  down  the  body  of  the  bolt  so  that  its  cross-sectional 
area  is  equal  to  the  area  at  the  root  of  the  thread ;  then  since  the 
total  elongation  of  the  bolt  depends  upon  the  length  of  this  re- 
duced section  it  follows  that  the  length  of  the  latter  should  be 
made  as  great  as  possible.  Such  a  bolt  is  weak  in  resisting  tor- 
sion and  flexure,  and  instead  of  fitting  the  hole  throughout  its 


ART.  85]  STRESSES  IN  SCREWS  99 

length,  it  merely  fits  at  the  points  where  the  body  was  not  turned 
down,  as  shown  in  Fig.  32(a).  Low  cost  of  production  is  the 
chief  advantage. 

The  tie  rods  used  in  bridge  and  structural  work  are  generally 
very  long,  and  the  prevailing  practice  calls  for  upset  threaded 
ends,  which  is  merely  another  way  of  making  the  cross-section  of 
the  body  of  the  rod  practically  the  same  as  the  area  at  the  root 
of  the  thread.  No  doubt  in  this  class  of  work  the  object  of 
making  the  rods  as  thus  described  is  to  save  weight  and  mater- 
ial; however,  it  should  be  pointed  out  that  the  capacity  for 
resisting  shocks  has  also  at  the  same  time  been  increased. 

2.  Instead  of  turning  down  the  body,  the  cross-sectional  area 
may  be  reduced  by  drilling  a  hole  from  the  head  of  the  bolt  to- 
ward the  threaded  end,  as  shown  in  Fig.  32(&).  This  method 
no  doubt  is  the  best,  as  the  bolt  fits  the  hole  throughout  its  length, 
and  the  hollow  section  is  well-adapted  to  resist  flexural  as  well 
as  torsional  stresses.  The  cost  of  production  may  be  excessive 
for  long  bolts  and  for  the  latter  the  method  of  Fig.  32 (a)  may  be 
employed. 

Actual  tests  were  made  by  Prof.  R.  C.  Carpenter  at  the  Sibley 
College  Laboratory  on  bolts  1J^  inches  in  diameter  and  12  inches 
long,  half  of  which  were  solid  and  the  remainder  had  their 
bodies  reduced  in  area  by  drilling  a  hole  as  shown  in  Fig.  32(6). 
Two  of  these  bolts,  tested  to  destruction,  showed  that  the  solid 
or  undrilled  bolt  broke  in  the  thread  with  a  total  elongation  of 
0.25  inch.  Additional  tests  in  which  similar  bolts  were  subjected 
to  shock  gave  similar  results. 

85.  Stresses  due  to  Combined  Loads. — Having  discussed  the 
individual  stresses  induced  in  bolts  and  screws  by  screwing  them 
up  and  by  the  external  loads  coming  upon  them,  it  is  in  order 
next  to  determine  the  stress  induced  by  the  combined  action  of 
these  loads.  This  resultant  stress  depends  upon  the  rigidity  of 
the  parts  fastened  as  well  as  upon  the  rigidity  of  the  screw  itself. 

(a)  Flanged  joint  with  gasket. — In  general  it  may  be  said  that 
for  an  unyielding  or  rigid  bolt  or  screw  fastening  two  machine 
parts  that  will  yield  due  to  screwing  up,  the  stress  in  the  bolt  is 
that  due  to  the  sum  of  the  initial  tension  due  to  screwing  up  and 
the  external  load,  as  the  following  analysis  will  show.  In  actual 
fastenings  used  for  machine  parts,  neither  the  bolt  nor  the  parts 
fastened  fulfill  the  above  conditions  absolutely;  however,  the 
conditions  are  very  nearly  approached  when  some  semi-elastic 


100 


STRESSES  IN  SCREWS 


[CHAP.  IV 


material  like  sheet  packing  is  used  to  make  a  tight  joint,  as  in 
steam  and  air  piping.  In  a  joint  such  as  illustrated  in  Fig.  33 (a) 
the  packing  acts  like  a  spring,  and  tightening  the  nut  will  com- 
press the  packing  a  small  amount,  thus  causing  a  stress  in  the 
bolt  corresponding  to  this  compression.  Assuming  that  an  ex- 
ternal load  due  to  some  fluid  pressure  acts  upon  the  flange  a,  its 
effect  will  be  to  elongate  the  stud  thereby  increasing  the  stress, 
and  at  the  same  time  reduce  slightly  the  pressure  exerted  upon 
the  stud  by  the  packing;  hence,  it  follows  that  for  this  case,  the 
load  upon  the  stud  may  for  all  practical  purposes  be  considered 
as  equivalent  to  the  sum  of  the  two  loads. 

(b)  Flanged  joint  without  gasket. — The  next  case  to  be  con- 
sidered is  that  type  of  fastening  in  which  the  stud,  bolt  or  screw 


(a) 


(b) 


FIG.  33. 


yields  far  more  than  the  connected  parts.  This  case  is  repre- 
sented by  two  flanges  having  a  ground  joint,  as  shown  in  Fig. 
33(6).  Due  to  screwing  up  of  this  joint,  the  stud  which  now 
elongates,  in  other  words  acts  like  a  spring,  will  be  subjected  to  a 
stress  corresponding  to  this  elongation.  If,  as  in  the  preceding 
case,  we  now  introduce  a  pressure  upon  the  flange  b  which  tends 
to  pull  the  fastening  apart,  it  is  evident  that  the  resultant  pres- 
sure at  the  ground  joint  is  the  difference  between  the  pressures 
exerted  by  the  nut  c  upon  the  outside  of  the  flange  b  and  that  due 
to  the  fluid  pressure  on  the  inside  of  b.  As  long  as  the  pressure 
on  the  inside  of  b  does  not  exceed  that  due  to  the  screwing  up  of 
the  nut,  the  stud  will  remain  unchanged  in  length;  hence  the 
stress  induced  is  that  due  to  the  initial  tension  and  not  that  due 
to  the  external  load.  If,  however,  the  pressure  on  the  inside  is 


ART.  86] 


STRESSES  IN  SCREWS 


101 


sufficient  to  overcome  that  due  to  the  nut,  the  joint  will  separate, 
causing  the  stud  to  elongate ;  hence  the  stress  in  the  latter  is  that 
due  to  the  external  load. 

86.  Fastening  with  Eccentric  Loading. — (a)  Rectangular  base. 
— In  Fig.  34  (a)  is  shown  the  column  of  a  drill  press  bolted  to 
the  cast-iron  base  by  cap  screws.  Due  to  the  thrust  P  of  the 
drill  which  tends  to  overturn  the  column,  these  screws  are  sub- 
jected to  a  tensile  stress  which  is  not  the  same  for  each  screw, 
as  the  analysis  below  will  show. 

To  determine  the  maximum  load  that  may  come  upon  any 
screw  we  shall  assume  that  the  column,  which  is  rigid,  is  fastened 


ll  — 


t 


FIG.  34. 

to  the  rigid  base  by  means  of  eight  screws,  as  shown  in  Fig.  34(6). 
Due  to  the  thrust  P,  the  column  will  tip  backward  about  the 
point  A,  thus  stretching  each  screw  a  small  amount  depending 
upon  its  distance  from  the  axis  A B,  Fig.  34(6).  Since  the  stresses 
induced  in  the  screws  vary  directly  as  the  elongations,  it  is 
evident  that  the  loads  upon  the  screws  vary. 

Now  the  moment  of  the  thrust  P  must  be  balanced  by  the  sum 
of  the  moments  of  the  screw  loads  about  the  axis  AB;  hence, 
representing  the  loads  upon  the  screws  by  Qi,  Q2>  etc.,  and  their 
moment  arms  by  l\t  Z2,  etc.,  it  follows  that 


PL  =  2(Q,Zi  +  Q2/2  +  Qzh  + 


(83) 


The  subscripts  used  correspond  to  the  number  of  the  screw 
as  shown  in  Fig.  34(6).     Since  the  stresses  induced  in  any  screw 


102 


STRESSES  IN  SCREWS 


[CHAP.  IV 


vary  directly  as  the  elongation  produced,  we  obtain  the  fol- 
lowing relations : 

f\  f\  f\  t~\  f\  f\  fQA\ 

(^2    ==    vfcl  T~  HJ3    —    «^1  T"  <^4    ==    «£l  y~  lorr) 

Substituting  these  values  in  (83),  the  expression  for  the  exter- 
nal moment  becomes: 

o  r\    .-  -• 

(85) 


From  the  preceding  discussion,  it  is  apparent  that  the  maxi- 
mum stresses  occur  in  the  screws  labeled  4,  and  the  magnitude 

of  this  maximum  stress  is  given  by 
A  the  following  expression: 

PLl<  (86) 


Knowing  the  various  dimensions, 
as  well  as  the  thrust  P,  the  magnitude 
of  Q4  is  readily  determined,  and  from 
this  the  size  of  screw  for  any  allow- 
able fiber  stress. 

(b)  Circular  base. — Instead  of  hav- 
ing a  rectangular  base  as  discussed 
above,  columns  or  machine  members 
are  frequently  made  with  a  circular 
base  similar  to  that  shown  in  Fig.  35, 
in  which  2a  represents  the  outside 
diameter  of  the  column  flange,  and 
26  the  diameter  of  the  bolt  circle.  For  the  case  under  dis- 
cussion six  bolts  or  screws  numbered  from  1  to  6  inclusive  are 
used.  Adopting  a  notation  similar  to  that  used  in  the  preceding 
analysis,  we  have  that  the  external  moment  due  to  the  load  P  is 


, .(..,___         ^ 

«--  ~"' . 

-  -  -  _ 


FIG.  35. 


From  the  geometry  of  the  figure 

11  =  a  —  b  cos  a 

12  =  a  -  b  cos  (60  +  a) 

13  =  a  +  b  cos  (60  -  a) 
li  =  a  -\-  b  cos  a 

h  =  a  +  b  cos  (60  +  a) 
16  =  a  —  b  cos  (60  —  a) 


(87) 


(88) 


ART.  87]  STRESSES  IN  SCREWS 

Substituting  these  values  in  (87),  it  follows  that 


103 


3  62  ~| 
os  oJ 


a  —  b  cos 

from  which  the  magnitude  of  Qi  is  given  by  the  following  expres- 
sion: 

Q.  =  PL[a~tb C°~]  (89) 

Now  to  determine  the  maximum  value  of  Qi  for  a  given  mo- 
ment PL  and  dimensions  a  and  b,  it  is  evident  from  (89)  that  this 
occurs  when  cos  a  is  a  minimum,  i.e.,  cos  a.  =  —  1,  which  is  the 
case  when  the  angle  a  is  180  degrees.  Hence 

PLr  a  +  b  1 


Knowing  the  maximum  load,  the  size  of  the  bolts  or  screws 
must  be  proportioned  for  this  load. 


FIG.  36. 

By  means  of  an  analysis  similar  to  the  above,  the  stresses  in 
any  number  of  bolts  or  screws  may  be  arrived  at. 

87.  Common  Bearing. — In  machinery,  many  forms  of  fasten- 
ings are  used  in  which  the  bolts  or  screws  are  subjected  to  shear- 
ing stresses  in  addition  to  tensile  stresses.  A  very  simple  form 
of  such  a  fastening  is  shown  in  Fig.  36,  which  represents  a  solid 
cast-iron  flanged  bearing  frequently  found  on  heavy  machine 
tools.  Due  to  the  power  transmitted  by  the  gears  located  on  the 
shaft,  the  bearing  is  subjected  to  a  pressure  P  which  tends  to 


104  POWER  SCREWS  [CHAP.  IV 

produce  a  shearing  stress  in  each  of  the  screws.  For  convenience, 
all  of  the  screws  are  assumed  to  be  stressed  equally.  As  men- 
tioned in  Art.  83,  dowel  pins  may  be  used  as  shown  in  Fig.  36, 
and  if  these  are  fitted  correctly  they  will,  to  a  great  extent  if  not 
altogether,  relieve  the  screws  from  a  shearing  action. 

Due  to  the  eccentric  location  of  P,  relative  to  the  supporting 
frame,  the  bearing  is  subjected  to  an  external  moment  PL, 
which  must  be  balanced  by  an  equal  moment  due  to  the -tension 
set  up  in  the  screws.  For  the  bearing  shown  in  Fig.  36  having  six 
screws  on  a  bolt  circle  of  diameter  26,  the  relation  between  the 
external  moment  and  the  moment  of  the  screw  loads  may  be 
obtained  from  (90). 

Now  assume  a  diameter  of  screw,  and  determine  the  direct 
shearing  stress,  if  no  dowel  pins  are  used,  also  the  tensile  stress 
caused  by  the  external  moment.  To  arrive  at  the  maximum  in- 
tensity of  stress,  combine  the  two  separate  stresses  by  means  of 
(28);  the  result  should  not  exceed  the  assumed  safe  working 
stress. 

POWER    SCREWS 

Three  forms  of  threads  adapted  to  the  transmission  of  power 
are  shown  in  Art.  76;  of  these  the  square  thread  is  looked  upon 
with  the  greatest  favor  due  to  its  higher  efficiency.  Instead  of 
having  single-threaded  screws,  it  is  not  unusual  to  employ  screws 
having  multiple  threads,  an  example  of  which  is  shown  in  the 
friction  spindle  press  illustrated  in  Fig.  125.  In  connection  with 
multiple-threaded  screws,  attention  is  called  to  the  terms  lead 
and  divided  pitch.  By  the  former  is  meant  the  distance  that  the 
nut  advances  for  one  revolution  of  the  screw,  and  by  the  latter, 
the  distance  between  consecutive  threads ;  hence  a  triple-threaded 
screw  of  one  and  one-half  inch  lead  has  a  divided  pitch  of  one- 
half  inch. 

88.  Efficiency  of  Square  Threads. — Referring  to  Fig.  37,  let  d 
represent  the  mean  diameter  of  the  screw.  The  action  of  the 
thread  upon  the  nut  is  very  similar  to  the  action  of  a  flat  pivot 
upon  its  bearing,  and  hence  we  shall  assume  that  the  pressure 
between  the  screw  and  the  nut  may  be  considered  as  concen- 
trated at  the  mean  circumference  of  the  thread. 

(a)  Direct  motion. — Representing  the  average  intensity  of 
pressure  between  the  screw  and  its  nut  by  the  symbol  q,  we  get 
for  the  total  pressure  on  a  small  area  6 A  of  the  surface  of  the 


ART.  88] 


EFFICIENCY  OF  SQUARE  THREADS 


105 


thread  qdA.  If  the  screw  is  rotated  so  that  the  axial  load 
Q  is  raised,  as  for  example  in  a  screw  jack,  the  pressure  qdA  will 
act  along  the  line  OB  making  an  angle  <p'  with  the  normal  OA. 
The  symbol  <?'  represents  the  angle  of  friction  for  the  surfaces  in 
contact.  Now  since  the  normal  OA  is  inclined  to  the  axis  of  the 
screw  by  the  angle  a,  the  angle  of  rise  of  the  mean  helix,  it  is 
evident  that  the  components  of  the  pressure  q8A  parallel  to  the 
axis  and  at  right  angles  thereto,  are  as  follows : 


Parallel    component        =  qdA  cos  (a  +  <?') 
Right  angle  component  =  qdA  sin  (a  +  <p') 

Hence  Q  =  q  cos  (a  +  <p')  2dA 

or  Q  =  qA  cos  (a  +  <p'),  (91) 

in  which  A  represents  the  total  surface  of  the  thread  in  actual 
contact. 

The  torsional  moment  of  the  component  at  right  angles  to  the 
axis,  about  the  axis,  is 

,m        qddA    .    f  ,. 

dT  =  ^— —  sin  (a  +  iff) 

Summing  up  for  the  entire  surface  in  contact, 

T  =  2^  sin  (a  +  *>')  (92) 

Since  q  and  A  are  generally  unknown  quantities,  it  is  desirable 


106  EFFICIENCY  OF  SQUARE  THREADS         [CHAP.  IV 

to  derive  an  expression  for  T  in  terms  of  the  load  Q.     This  may 
be  done  by  combining  (91)  and  (92),  whence 

T  =  0-  tan  (a  +  <p')  (96) 

In  order  to  obtain  an  expression  for  the  efficiency  of  the  square- 
threaded  screw,  determine  the  torsional  moment  To  required 
when  friction  is  not  considered,  and  divide  this  moment  by  T. 
Without  friction  <p'  =  0;  hence,  from  (93),  it  follows  that 

TQ  =  Y  tan  «  (94) 

Hence,  the  efficiency  is 


-  - 

'         ~ 


~  tan  (a  +  /) 

The  expression  given  by  (95)  could  have  been  obtained  directly 
from  (79)  by  making  j8  =  90°. 

Very  often  it  is  more  desirable  to  have  the  expressions  for  T 
and  i\  in  terms  of  the  coefficient  of  friction  and  the  dimensions  of 
the  screw.  Letting  p  represent  the  pitch  of  the  screw,  then 

T) 

tan  a  =  A;  also  tan  <p'  =  //.     Substituting  these  values  in  (93) 
and  (95),  the  resulting  expression  for  T  is 

Qdrp  +  w'di  , 

TlS^T^J 

and  that  for  the  efficiency  is 

irdp  -  n'p*  . 

71  ~  vdp  +  TT  W2 

The  value  of  the  coefficient  of  friction  varies  greatly  with  the 
method  of  lubrication  and  the  quality  of  the  lubricant.  Very 
little  experimental  information  on  threads  is  available  ;  probably 
the  most  reliable  being  the  results  obtained  by  Prof.  A.  Kingsbury 
from  an  extended  series  of  tests  on  square-threaded  screws  made 
of  various  materials,  such  as  mild  steel,  case-hardened  mild 
steel,  wrought  iron,  cast  iron,  and  cast  bronze.  The  results  of 
this  investigation,  some  of  which  are  given  in  Table  24,  were 
presented  in  a  paper  before  the  American  Society  of  Mechanical 
Engineers  by  Prof.  Kingsbury,  and  form  a  part  of  volume  17  of 
the  Transactions  of  that  society.  In  the  second  last  column  of 
Table  24  are  given  the  mean  values  of  the  coefficient  of  friction 


ART.  88]  EFFICIENCY  OF  SQUARE  THREADS  107 

TABLE  24. — COEFFICIENTS  OF  FRICTION  FOR  SQUARE-THREADED  SCREWS 


Material    used    for 
screw 

Pres. 
per 
sq.   in. 

Material   used   for   nut 

Aver, 
coef. 

Lubricant 
used 

Cast 
iron 

Wrought 
iron 

Mild 
steel 

Cast 
brass 

Cast  iron  
Wrought  iron  

3,000 

0.1400 
0.1500 
0.1320 
0.1675 
0.1300 

0.1570 
0.1600 
0.1560 
0.1775 
0.1300 

0.1500 
0.1500 
0.1470 
0.1550 
0.1270 

0.1200 
0.1170 
0.1270 
0.1325 
0.1400 

0.143 

Heavy 
machinery 
Oil 

Mild  steel  
Mild  steel  case-hard... 
Cast  bronze  

Cast  iron  
Wrought  iron  
Mild  steel  
Mild  steel  case-hard... 
Cast  bronze  

10,000 

0.1190 
0.1380 
0.1360 
0.1300 
0.1720 

0.1390 
0.1400 
0.1600 
0.1430 
0.1350 

0.1250 
0.1390 
0.1410 
0.1330 
0.1240 

0.1710 
0.1470 
0.1360 
0.1930 
0.1320 

Cast  iron                  .... 

10,000 

0.1050 
0.0750 
0.0650 
0.0700 
0.0440 

0.0710 
0.0700 
0.0675 
0.0550 
0.0450 

0.1075 
0.0890 
0.1110 
0.1275 
0.0710 

0  .  0590 
0.0550 
0.0400 
0.0350 
0.0360 

0.07 

Heavy 
machinery 
oil 
and 
graphite 

Wrought  iron  
Mild  steel          

Mild  steel  case-hard... 
Cast  bronze  

10,000 

0  .  0950 
0.1000 
0.1000 
0.1050 
0.1100 

0.1000 
0.1075 
0.1050 
0.0975 
0.1000 

0.1000 
0.1125 
0.1200 
0.1175 
0.1150 

0.1100 
0.1200 
0.1100 
0.1375 
0.1325 

0.11 

Lard   oil 

Wrought  iron  
Mild  steel        

Mild  steel  case-hard... 
Cast  bronze  

for  the  various  lubricants  as  determined  by  Prof.  Kingsbury, 
and  these  values  are  applicable  to  square-threaded  screws  run- 
ning at  very  slow  speeds  and  upon  which  the  bearing  pressure 
does  not  exceed  14,000  pounds  per  square  inch,  provided  the 
screw  is  lubricated  freely  before  the  pressure  is  applied. 

(b)  Reverse  motion. — For  the  reverse  motion  of  the  screw,  the 
line  of  action  of  the  pressure  qdA  is  inclined  to  the  axis  of  the 
screw  at  the  angle  (a  —  (pf) ;  hence  the  moment  required  to  turn 
the  screw  is 


(D  =       tan  («-*, 

_  Qdrp  -  irpi'd-1 
2  Lird  +  n'p\ 


(98) 


For  the  common  screw  jack  and  screws  for  elevating  the  cross 
rail  on  planers,  boring  mills,  and  large  milling  machines,  the  angle 
of  friction  <pf  exceeds  the  angle. «,  thus  making  (T)  negative;  that 
is,  the  lowering  of  the  load  Q  requires  an  effort  or  in  other  words, 
the  screw  is  said  to  be  self-locking.  If,  however,  <p'  <  a,  the 
moment  is  positive ;  that  is,  an  effort  must  be  applied  to  resist  the 
tendency  of  the  load  to  descend. 


108 


STRESSES  IN  POWER  SCREWS 


[CHAP.  IV 


From  the  above  discussion,  it  is  evident  that  in  a  self-locking 
screw,  the  limiting  value  of  a  is  <pr.  Substituting  a  =  v'  in  (95)  , 
the  maximum  efficiency  of  a  self-locking  screw  is 


-n  = 


that  is  to  say  v\  in  this  case  can  never  exceed  50  per  cent. 

89.  Stresses  in  Power  Screws.  —  Screws  used  for  the  trans- 
mission of  power  are  subjected  to  the  following  stresses:  bearing, 
tensile  or  compressive,  and  shearing. 

(a)  Bearing  stresses.  —  In  order  that  the  thread  of  a  screw  may 
be  capable  of  transmitting  the  required  power  without  an  undue 
amount  of  wear,  the  unit  pressure  upon  the  surfaces  in  contact 
must  be  kept  low,  especially  if  the  rubbing  speeds  are  high.  In- 
stead of  giving  this  permissible  pressure  in  terms  of  the  normal 
pressure  per  square  inch  of  actual  contact,  it  is  generally  quoted 
as  so  many  pounds  per  square  inch  of  projected  area.  To  deter- 
mine an  expression  for  this  quantity  in  terms  of  the  load  on  the 
screw,  proceed  as  follows:  Using  the  notation  of  Art.  88,  the 
projected  area  of  the  total  thread  surface  in  actual  contact  be- 

tween the  nut  and  its  screw  is  —r  (d\  —  dl),  hence 


Q  = 


(100) 


in  which  n  and  Sb  represent  the  number  of  threads  in  contact  and 
the  permissible  pressure  per  square  inch  of  projected  area, 
respectively. 

The  values  of  Sb  given  in  Table  25  were  determined  from 
actual  screws  in  service,  and  may  serve  as  a  guide  in  future 
calculations. 

TABLE  25.  —  BEARING  PRESSURES  ON  POWER  SCREWS 


Service 

Material 

Bearing  pressures 

Remarks 

Screw 

Nut 

Min. 

Max. 

Mean 

Jack  screw  
Hoisting  screw  . 
Hoisting  screw. 

Steel 
Steel 
Steel 

Cast  iron 
Cast  Iron 
Brass 

1,800 
500 
800 

2,600 
1,000 
1,400 

2,200 
750 
1,100 

Slow  speed 
Medium  speed 
Medium  speed 

(b)  Tensile  or  compressive  stresses. — The  method  of  mounting 
the  screw,  and  the  manner  of  transmitting  the  desired  power, 


ART.  89]  STRESSES  IN  POWER  SCREWS  109 

determine  the  kind  of  stress  induced  in  the  screw  by  the  action  of 
the  direct  load.  The  magnitude  of  this  stress  is  equivalent  to 
the  load  divided  by  the  area  at  the  root  of  the  thread,  provided 
the  length  of  the  screw  if  subjected  to  compression  does  not  ex- 
ceed six  or  eight  times  the  root  diameter.  If  a  screw  subjected 
to  a  compression  has  a  length  exceeding  the  limits  just  given, 
it  must  be  treated  as  a  column,  and  the  stresses  determined  ac- 
cording to  the  formulas  given  in  Art.  15.  It  is  good  practice  to 
neglect  any  stiffening  effect  that  the  threads  may  have. 

(c)  Shearing  stresses.  —  A  torsional  or  shearing  stress  is  induced 
in  the  screw  by  the  external  turning  moment  applied,  though  a 
part  of  the  latter  may  also  be  used  in  overcoming  the  friction  of 
bearings,  depending  upon  the  arrangement  of  the  screw  and  nut. 
In  general,  the  magnitude  of  the  moment  causing  the  shearing  is 
never  less  than  that  given  by  (93)  or  (96),  and  hence  the  shearing 
stress  induced  in  this  case  is 

&  =  (101) 


(d)  Combined  stresses.  —  Having  determined  the  magnitude  of 
the  separate  stresses  induced  in  the  screw,  their  combined  effect 
must  be  determined  by  the  principles  explained  in  Art.  17. 


CHAPTER  V 


FASTENINGS 

KEYS,  COTTERS,  AND  PINS 
KEYS 

The  principal  function  of  keys  and  pins  is  to  prevent  relative 
rotary  motion  between  two  parts  of  a  machine,  as  of  a  pulley 
about  a  shaft  on  which  it  fits.  In  general,  keys  are  made  either 
straight  or  slightly  tapering.  The  straight  keys  are  to  be  pre- 
ferred since  they  will  not  disturb  the  alignment  of  the  parts 
to  be  keyed,  but  have  the  disadvantage  that  they  require  accu- 
rate fitting  between  the  hub  and  shaft.  The  taper  keys  by  taking 
up  the  slight  play  between  the  hub  and  shaft  are  likely  to  throw 


FIG.  38. 

the  wheels  or  gears  out  of  alignment,  but  they  have  the  advan- 
tage that  any  axial  motion  between  the  parts  is  prevented  due  to 
the  wedging  action.  Keys  may  be  divided  into  three  classes  as 
follows:  (a)  sunk  keys;  (b)  keys  on  flats;  (c)  friction  keys. 

90.  Sunk  Keys. — The  types  of  sunk  keys  used  most  in  machine 
construction  are  those  having  a  rectangular  cross-section,  though 
occasionally  round  or  pin  keys  are  used. 

(a)  Square  key. — The  so-called  square  key  is  only  approxi- 
mately square  in  cross-section  and  has  its  opposite  sides  parallel. 
As  shown  in  Fig.  38 (a),  this  type  of  key  bears  only  on  the  sides 
of  the  key  seats,  and,  being  provided  with  a  slight  clearance  at 
the  top  and  bottom,  the  key  has  no  tendency  to  exert  a  bursting 

110 


ART.  90] 


SUNK  KEYS 


111 


pressure  upon  the  hub.  To  prevent  axial  movement  of  the  hub, 
set  screws  bearing  upon  the  key,  or  other  means  must  be  pro- 
vided. The  square  key  is  used  where  accurate  concentricity  of 
the  keyed  parts  is  required,  also  when  the  parts  must  be  dis- 
connected frequently,  as  in  machine  tools.  It  is  suitable  for 
heavy  loads,  provided  set  screws  are  used  to  prevent  tipping  of 
the  key  in  its  seat.  For  a  list  of  commercial  sizes  of  square  keys 
see  Table  29  and  Fig.  45 (a),  to  which  the  dimensions  in  the 
table  refer. 

(b)  Flat  key. — The  flat  key  has  parallel  sides,  but  its  top  and 
bottom  taper.  As  shown  in  Fig.  38(6),  its  thickness  t  is  consider- 
ably less  than  its  width  b;  furthermore,  it  fits  on  all  sides,  thus 
tending  to  spring  the  connected  parts  and  at  the  same  time 
introducing  a  bursting  pressure  upon  the  hub.  The  flat  key  is 
used  for  either  heavy  or  light  service  in  which  the  objections  iust 
mentioned  are  not  serious. 


(a) 


(b) 


FIG.  39. 


(c)  Feather  key. — The  feather  key,  sometimes  called  spline,  is 
a  key  fitted  only  on  the  sides,  thus  permitting  free  axial  move- 
ment of  the  hub  along  the  shaft.     Its  thickness  is  usually  greater 
than  its  width,  thereby  increasing  the  contact  surface  and  at  the 
same  time  decreasing  the  wear.     The  feather  key  is  fastened  to 
either  the  hub  or  the  shaft,  while  the  key-way  in  the  other  part 
is  made  a  nice  sliding  fit.     The  key  may  be  secured  to  the  shaft 
by   countersunk   machine   screws  or   by  pins  riveted  over;  or 
when  it  is  desired  to  fasten  the  key  to  the  sliding  hub,  dovetailing 
or  riveting  may  be  resorted  to.     Quite  frequently  two  feather 
keys  set  180  degrees  apart  are  used.     The  stresses  are  thereby 
equalized,  and  at  the  same  time  it  is  easier  to  slide  the  hub  along 
the  shaft. 

(d)  Woodruff  key. — The  Woodruff  key  shown  in  Fig.  39 (a)  is 
a  modified  form  of  the  sunk  key.     It  is  patented  and  is  manu- 
factured by  the  Whitney  Mfg.  Co.  of  Hartford,  Conn.     The  key- 


112  DIMENSIONS  OF  WOODRUFF  KEYS  CHAP.  V 

TABLE  26. — DIMENSIONS  OF  WOODRUFF  KEYS 


No. 

i 

2 

3 

4 

Key 

length 

No. 

i 

2 

3 

4 

Key 
length 

1 

2 
3 

•X 

K\ 

X 

H. 

X 

23 
F 

IK 

KG 

H2 

IH 

24 
25 
G 

.« 

KG 

X. 

IK 

4 
5 
6 
61 

„ 

3/32 

X 

%2 

y^ 

r. 

126 
127 
128 
129 

« 

eo\  i-K  iOX  co\ 

.,, 

%2 

i»«, 

7 
8 
9 
91 

H 

X 

til 

1A* 

% 

26 
27 

28 
29 

« 

tO  50 

\--i  X:*  \>-H  \oo 

COX  rH\  HSS  COX 

l7/32 

Hi 

10 
11 

12 
A 

H 

<N  (0  IN 

X«  V-i  \eo  V* 

U5\  COX  t-\  r-f\ 

1A* 

« 

~R.^ 

Sx 
Tx 
Ux 
Vx 

2% 

K 

S6 

KG 

25/32 

0.1625 

2 

13 
14 
15 
B 
152 

i 

KG 

K2 
KG 

lAe 

1 

R 

S 
T 
U 
V 

2% 

%6 
H 

K 

H 

2^6 

16 
17 

18 
C 

* 

KG 
KG 

,, 

* 

30 
31 
32 
33 
34 
35 

36 

« 

% 

H 

,. 

KG 

2% 

19 
20 
21 
D 
E 

* 

KG 
KG 

,, 

« 

22 

m 

X 

%* 

m 

seat  in  the  hub  is  of  the  usual  form,  but  that  in  the  shaft  has  a 
circular  outline  and  is  considerably  deeper  than  the  ordinary 
key-way.  The  extra  depth,  of  course,  weakens  the  shaft,  but  the 
deep  base  of  the  key  precludes  all  possibility  of  tipping. 

The  freedom  of  the  key  to  adjust  itself  to  the  key-seat  in  the 
hub  makes  an  imperfect  fit  almost  impossible,  while  with  the  ordi- 
nary taper  key  a  perfect  fit  is  very  difficult  to  obtain.  In  secur- 


ART.  90] 


EARTH  KEY 


113 


ing  long  hubs,  the  depths  of  the  key-way  may  be  diminished  by 
using  two  or  more  Woodruff  keys  at  intervals  in  the  same  key- 
seat. 

In  Table  26  are  given  the  stock  sizes  of  Woodruff  keys,  also 
the  various  dimensions  referred  to  in  Fig.  40. 

To  aid  the  designer  in  selecting  the  suitable  size  of  Woodruff 
key  for  any  given  diameter  of  shaft,  the  information  contained 
in  Table  27  may  be  found  convenient. 


(b) 


FIG.  40. 


TABLE  27. — DIAMETERS  OF  SHAFTS  AND  SUITABLE  WOODRUFF  KEYS 


Shaft 
diam. 

Key 
No. 

Shaft 
diam. 

Key 
No. 

Shaft 
diam. 

Key 

No. 

5A6-y8 

1 

%-    15/1G 

6,    8,10 

1%    -IKe 

14,  17,  20 

7/i6-y2 

2,4 

1 

9,  11,  13 

1H-1K 

15,  18,  21,  24 

K*-5A 

3,5 

iMe-lH 

9,  11,  13,  16 

iiyi6-iH 

18,  21,  24 

lMe-H 

3,5,7 

i«e 

11,  13,  16 

1^6-2 

23,  25 

lHe 

6,8 

1M   ~1%6 

12,  14,  17,  20 

2Ke  -2K 

25 

(e)  Lewis  key. — The  type  of  sunk  key  shown  in  Fig.  39(6)  was 
invented  by  Mr.  Wilfred  Lewis.  This  key  is  subjected  practi- 
cally to  a  pure  compression  in  the  direction  of  its  longest  cross- 
sectional  dimension,  and  for  that  reason  the  location  of  this  key 
relative  to  the  direction  of  driving  is  very  important.  The  Lewis 
key  is  rather  expensive  to  fit  and  probably  due  to  that  fact  is  not 
used  so  extensively,  though  at  the  present  time  one  manufacturer 
uses  it  on  large  engine  shafts.  Frequently  two  such  keys  are 
used  on  one  hub. 

(/)  Earth  key. — Some  years  ago  Mr.  C.  G.  Earth  invented  the 
type  of  key  shown  in  Fig.  41  (a).  It  consists  of  an  ordinary 
rectangular  key  with  one-half  of  both  sides  beveled  off  at  45 
degrees.  With,  this  form  of  key  it  is  not  necessary  to  make  a 
tight  fit,  since  the  pressure  tends  to  force  the  key  into  its  seat. 


114 


KENNEDY  KEYS 


[CHAP.  V 


Furthermore,  there  is  no  tendency  for  the  key  to  turn  in  its  seat, 
since  the  pressure  upon  it  produces  a  compression.  With  re- 
spect to  the  stresses  produced,  this  key  is  similar  to  the  Lewis 
key,  but  has  the  advantage  over  the  latter  that  it  costs  less  to 
fit.  The  Earth  key  may  also  be  used  as  a  feather  key;  in  many 
cases  it  has  replaced  troublesome  rectangular  feather  keys  and 
has  always  given  excellent  service. 


(b) 


FIG.  41. 


(g)  Kennedy  keys.— Another  system  of  keying,  which  has 
given  excellent  service  in  heavy  rolling-mill  work,  is  shown  in  Fig. 
41  (b) .  This  system,  known  as  the  Kennedy  keys,  is  similar  to  that 
in  which  two  Lewis  keys  are  used  in  one  hub.  The  two  keys  are 
located  in  the  hub  in  such  a  manner  that  the  diagonals  pass 
through  the  center  of  the  shaft  as  shown  in  the  figure.  The 
dimensions  of  the  key  at  the  smaller  end  are  made  approximately 
one-fourth  of  the  diameter  of  the  shaft,  and  the  taper  is  made 


FIG.  42. 

%  inch  per  foot.  The  key  should  form  a  driving  fit  at  the  top 
and  bottom.  The  following  method  of  fitting  a  hub  with  Ken- 
nedy keys  represents  the  practice  of  a  well-known  manufacturer, 
and  when  thus  fitted,  such  keys  have  always  given  good  results. 
1  'The  hub  of  the  gear  after  being  bored  for  a  press  fit  with  its 
shaft  is  rebored  by  offsetting  the  center  approximately  ^4  inch, 
thus  producing  the  clearance  shown  in  the  figure.  The  keys  are 
fitted  on  the  eccentric  side  of  the  bore  and  hence  when  driven 
home  pull  the  hub  into  its  proper  place."  The  reboring  opera- 


ART.  91] 


PIN  KEYS 


115 


tion  is  not  essential  to  insure  good  results,  but  it  facilitates 
erection  of  the  parts. 

(h)  Round  or  pin  key. — A  round  or  pin  key  gives  a  cheap  and 
accurate  means  of  securing  a  hub  to  the  end  of  a  shaft.  This 
form  of  fastening,  shown  in  Fig.  42 (a),  was  originally  intended 
only  for  light  and  small  work,  but  if  properly  designed  and  con- 
structed will  also  prove  satisfactory  for  heavy  work.  The  pin, 
either  cylindrical  or  tapering,  is  fitted  halfway  into  the  shaft  and 
hub  as  shown  in  the  figure.  For  heavy  duty,  the  Nordberg 
Mfg.  Co.  of  Milwaukee  uses  the  proportions  given  in  Table  28,  the 
total  taper  of  the  reamer  being  Jfg  inch  per  foot. 

For  light  duty  when  taper  pins  are  used,  it  is  advisable  to 
make  use  of  the  so-called  "  standard  taper  pins,"  as  they  may  be 
purchased  for  less  money  than  it  is  possible  to  make  them.  In 
Table  28  are  given  the  proportions  of  such  pins,  also  informa- 
tion pertaining  to  the  reamers  for  these  pins.  The  standard 
taper  is  y±  inch  per  foot. 


TABLE  28. — ROUND  KEYS  AND  TAPER  PINS 


Nordberg  round  keys 

Standard  taper  pins  and  reamers 

Reamer 

Pins 

Reamer 

Shaft 
diameter 

Small 
diam. 

Length 
of  flutes 

No. 

Large  diameter 

Stock 
lengths 

No. 

Small 
diam. 

Length 
of  flutes 

Actual 

Approx. 

2*^6-3 

H 

4>£ 

0 

0.156 

Kz 

?i-l?i 

0 

0.135 

1!K6 

3Ke-3^ 

H 

4H 

1 

0.172 

1H4. 

H-2 

1 

0.146 

IHe 

2H-4 

1    . 

4K 

2 

0.193 

X* 

X-2X 

2 

0.162 

1*^6 

4H-4M 

IX 

5 

3 

0.219 

Ha 

\ 

3 

0.183 

2H6 

5 

IK 

**A 

4 

0.250 

H 

\K-a 

4 

0.208 

2% 

SM 

m 

&A 

5 

0.289 

1H* 

} 

5 

0.240 

2H 

6 

\\i 

6« 

6 

0.341 

lHa 

K-4 

6 

0.279 

3H 

7,    8,    9 

IK 

6^,8 

7 

0.409 

X«2 

1     -4 

7 

0.331 

4K6 

10,  11,  12 

2 

lOtf 

8 

0.492 

H 

1K-4M 

8 

0.398 

5X 

13,  14,  15 

2^6 

12 

9 

0.591 

1K* 

1H-5K 

9 

0.482 

SK 

16,  17,  18 

3H 

13 

10 

0.706 

*Kt 

1M-6 

10 

0.581 

7 

19,  20,  21 

31K« 

22,  23,  24 

4tf 

14  K 

91.  Keys  on  Flats. — A  key  on  the  flat  of  a  shaft  has  parallel 
sides  with  its  top  and  bottom  slightly  tapering,  and  is  used  for 
transmitting  light  powers.  Fig.  42(6)  shows  this  form  of  fasten- 
ing. The  proportions  of  keys  on  flats  are  about  the  same  as 
those  used  for  the  flat  key  described  in  Art.  90(6). 


116 


STRENGTH  OF  KEYS 


[CHAP.  V 


92.  Friction  Keys. — The  most  common  form  of  friction  key  is 
the  saddle  key  shown  in  Fig.  43 (a),  the  sides  of  which  are  parallel, 
and  the  top  and  bottom,  slightly  tapering.     The  bottom  fits  the 
shaft  and  the  holding  power  of  the  key  is  due  to  friction  alone. 
This  form  of  key  is  intended  for  very  light  duty,  or  in  some  cases 
for  temporary  service,  as  in  setting  an  eccentric. 

93.  The  Strength  of  Keys. — Keys  are  generally  proportioned 
by  empirical  formulas,  and  in  almost  all  cases  such  formulas  are 
based  upon  the  diameter  of  the  shaft.     Neither  the  twisting 
moment  on  the  shaft  nor  the  length  of  the  key  is  considered  in 
arriving  at  the  cross-section.     Since   a  key  is  used  for  torsion 
alone,  the  twisting  moment  to  be  transmitted  and  not  the  diam- 
eter of  the  shaft  should  fix  its  dimensions.     In  the  majority 
of  cases  the  shaft  must  also  resist  a  bending  stress  in  addition 
to  the  torsional  stress,  and  a  larger  shaft  is  required  than  would 


(b) 


Fio.  43. 


be  necessary  for  simple  torsion.  The  empirical  formula  therefore 
give  a  larger  key  than  is  really  needed,  thereby  increasing  the 
cost  and  at  the  same  time  decreasing  the  effective  strength  of  the 
shaft.  The  length  of  the  key  should  be  considered  in  determining 
its  crushing  and  shearing  resistance. 

In  arriving  at  the  dimensions  of  the  key,  the  size  of  the  shaft 
should  not  be  disregarded  altogether,  or  the  result  might  be  a  key 
too  small  to  be  fitted  properly,  or  one  that  is  too  large.  In  other 
words,  calculate  the  dimensions  of  the  required  key  and  if  neces- 
sary modify  these  dimensions  to  suit  practical  considerations. 
It  is  generally  supposed  that  keys  fail  by  cross-shearing,  but  this 
is  seldom  the  case.  A  large  number  of  failures  are  due  to  the 
crushing  of  the  side  of  the  key  or  key-seat,  and  for  that  reason  the 
crushing  stress  should  always  be  investigated. 

(a)  Crushing  strength. — To  determine  the  crushing  stress  on 
the  side  of  a  key-seat,  let  T  represent  the  torsional  moment 


ART.  94]  STRENGTH  OF  KEYS  117 

transmitted,  I  the  length  of  the  key,  and  b  and  t,  the  dimensions 
indicated  in  Fig.  43(6).     Then   the  crushing  resistance  of  the 

//  Sf 
key  is  -^,  and  its  moment  about  the  center  of  the  shaft,  whose 

,77  rr 

diameter  is  d,  is  approximately      .    .     Equating  this  moment  to 
the  torsional  moment,  and  solving  for  $&,  we  have 

s>  =  w  (102) 

Assuming  $&  and  having  given  values  for  T,  t  and  d,  (102)  may 
be  used  for  calculating  the  required  length  of  the  key. 

Occasionally  a  key  is  required  to  transmit  the  full  power  of 
the  shaft;  hence,  making  its  strength  equal  to  that  of  the  shaft, 
we  get 


4  16 

from  which 

•-3S  <-> 

(b)  Shearing  strength.  —  The  shearing  stress  in  a  key  is  found 
by  equating  the  torsional  moment  T  to  the  product  of  the  radius 
of  the  shaft  and  the  stress  over  the  area  exposed  to  a  shear; 
whence 

s--m  (104) 

Equating  the  value  of  T  from  (102)  to  that  obtained  from  (104) 

t  =  2  b  ~  (105) 

kb 

If  Sb  =  2  Ss,  as  is  generally  assumed,  (105)  calls  for  a  square 
key.  To  facilitate  fitting,  the  width  of  the  key  is  frequently 
made  greater  than  its  depth,  which  has  the  effect  of  decreasing 
S8  relative  to  Sb.  From  this  it  follows  that  investigations  for 
the  crushing  stress  are  more  essential  than  those  for  the  shearing 
stress,  as  in  actual  practice  the  latter  takes  care  of  itself. 

94.  Friction  of  Feather  Keys.  —  As  stated  in  Art.  90(c),  it  is 
possible  to  equalize  the  pressure  coming  upon  the  hub  by  using 
two  feather  keys  placed  180  degrees  apart,  thereby  reducing 
materially  the  force  required  to  slide  the  hub  along  the  shaft. 
The  following  analysis  will  serve  to  show  that  the  statement 
is  practically  true. 


118 


FRICTION  OF  FEATHER  KEYS 


[CHAP.  V 


(a)  Hub  with  one  feather  key. — In  Fig.  44  (a)  is  shown  a  hub 
which  is  made  an  easy  sliding  fit  on  the  shaft  and  key,  the  latter 
being  fastened  securely  to  the  shaft.     We  shall  assume  that  the 
hub  drives  the  shaft  in  the  direction  indicated  by  the  arrow;  hence 
the  torsional  moment  T  transmitted  produces  the  two  forces 
PI,  one  of  which  acts  on  the  key  and  the  other,  having  the  same 
magnitude,  causes  a  pressure  on  the  shaft.     These  forces  being 
parallel  form  a  couple  whose  moment  Pitt  must  equal  the  tor- 
sional moment  T\  hence,  the  magnitude  of  the  force  PI  is 

Pi  =  -  (106) 

(b)  Hub  with  two  feather  keys. — In  place  of  a  single  feather, 
suppose  the  shaft  is  equipped  with  two  keys  upon  which  the  hub 
slides  as  shown  in  Fig.  44(6).     Assuming  the  direction  of  rotation 


(b) 


FIG.  44. 


shown  in  the  figure,  the  forces  upon  the  hub  are  the  two  equal 
forces  P2  forming  a  couple  whose  moment  is  2  P2a.  Since  the 
magnitude  of  this  couple  is  a  measure  of  the  torsional  moment  T, 
it  follows  that 

T 
Pz  =  2a  (107) 

Comparing  (106)  and  (107),  it  is  quite  evident  that  the  force 
producing  the  frictional  resistance  in  case  (6)  is  only  one-half 
as  great  as  that  in  case  (a),  assuming  the  same  values  of  T  and 
a,  thus  showing  the  advantages  gained  by  the  use  of  two  feather 
keys. 

It  is  important  to  note  that  the  hub  with  two  feather  keys 
requires  very  accurate  fitting  in  order  to  produce  the  action 
assumed  in  the  above  analysis. 


ART.  951 


GIB-HEAD  KEY 


119 


95.  Gib -head  Key. — The  gib-head  or  hook-head  key  is  shown 
in  Fig.  45,  and  is  nothing  more  than  a  flat  or  square  key  with  the 
head  added.  This  form  of  key  is  used  in  places  where  it  is  in- 
convenient or  practically  impossible  to  drive  out  a  key  from  the 
small  end.  It  should  be  borne  in  mind,  however,  that  a  project- 
ing head  is  always  a  source  of  danger  and  for  that  reason 
many  engineers  condemn  its  use.  In  Table  29  are  given  the 
dimensions  of  a  series  of  sizes  of  gib-head  keys  indicated  in  Fig. 


1 


(Q) 

FIG.  45. 

TABLE    29. — DIMENSIONS    OF    GIB-HEAD    KEYS 


K 
KG 

K 
KG 

X 

KG 

K 
KG 


% 
K 
1Kf 
1 

IK' 

1K6 

IK 

IKe 


K 


KG 

KG 

K 

KG 


1 
IKe 


We 


1 
IKe 

IKe 

IK  6 

IKe 

IK 

IHe 


KG 

15/B2 

KG 
% 


1 

IK 

We 

IK 

iHe 

IK 


IK 


2 

2K 


2K 
2% 


IK 
i1? 

IK 


2 
2K6 

2K 


2K6 


2K 

2K6 
2% 


2K 

2^e 
3 


IK 


IK 
1 

2 
2 

2K 


2 

2K6 

2K 

2K6 


2K 
2K6 


2K 
2K6 

2% 

2i  KG 

2K 

2^6 

3 


2K 


3 

3K6 

3K 


3Kc 
3K 


2K 
3 

3K 


3K 
4 

4K 


4K 


4K 

5 

5 

5K 

5K 


5% 


120  SHAFT  SPLINES  [CHAP.  V 

45 (a).     The  keys  listed  in  this  table  are  square  in  cross-section 
at  the  head  end,  and  have  a  taper  of  J^  inch  per  foot. 

96.  Key  Dimensions. — In  Fig.  45(6)  are  shown  the  dimensions 
that  will  prove  most  convenient  for  the  shop  man  in  order  to 
machine  the  key-seats  in  the  hub  and  shaft.     The  dimension  a 
is  the  one  used  for  arriving  at  the  proper  depth  of  the  key-seat 
in  the  hub.     To  arrive  at  the  depth  of  the  key-seat  in  the  shaft, 
the  majority  of  the  workmen  prefer  to  have  given  the  dimension 
c,  .as  that  is  by  far  the  most  convenient  dimension  when  the  key- 
seat  is  cut  on  a  milling  machine.     Some  mechanics  prefer  to  use 
the  dimension  e  in  place  of  c  thus  enabling  them  to  use  calipers. 

97.  Integral  Shaft  Splines. — With  the  development  of  the  auto- 
mobile, the  defects  of  the  inserted  keys  in  circular  shafts  became 
apparent,  and  finally  the  old  key  construction  was  discarded 
almost  altogether,  in  particular  in  the  sliding-gear  construction 
and  rear-axle  transmissions.     Due  to  the  weakening  of  the  shaft 
by  the  inserted  key,  the  square  shaft  was  at  first  introduced,  and 
this  met  with  considerable  success.     The  square  shaft,  however, 
is  considerably  heavier  than  a  circular  shaft  of  the  same  strength, 
so  in  order  to  keep  the  weight  down  and  at  the  same  time  provide 
greater  key-bearing  area,  the  automobile  designer  developed  what 
is  now  called  the  integral  spline  shaft.     Such  a  shaft  is  simply 
a  round  shaft  in  which  the  splines  are  produced  by  milling  out 
the  metal  between  them. 

At  first  the  integral  spline  shafts  were  produced  on  the  milling 
machine,  but  at  present  they  can  be  produced  more  cheaply  on 
the  hobbing  machine.  The  splined  holes  through  the  hubs  of  the 
gears  which  slide  over  such  shafts  are  produced  very  accurately 
and  cheaply  on  a  broaching  machine.  It  is  claimed  by  some 
manufacturers  that  the  cost  of  hobbing  a  multiple-spline  shaft 
and  broaching  the  hub  to  fit  the  shaft  is  considerably  less  than 
the  combined  cost  of  turning  the  circular  shaft,  cutting  the  key- 
way  in  it,  boring  the  gear  to  fit  the  shaft,  cutting  the  key-way  in 
the  gear,  and  fitting  the  key. 

The  automobile  manufacturer  is  not  the  only  one  that  is  using 
integral  spline  shafts ;  the  advantages  of  such  shafts  are  so  appar- 
ent that  a  considerable  number  of  machine  tool  builders  are  now 
using  them  in  connection  with  their  sliding  change-gear  mechan- 
isms. As  now  used  in  the  various  classes  of  service,  the  integral 
spline  shafts  are  constructed  with  from  four  to  ten  splines.  In 


ART.  97] 


SHAFT  SPLINES 


121 


Fig.  46  (a)  and  (b)  are  shown  the  cross-sections  of  a  hub  contain- 
ing six  and  ten  splines  respectively,  the  former  being  used  for 
the  sliding  gears,  while  the  latter  is  applied  to  the  rear  axle.  The 
proportions  of  the  two  types  shown  in  Fig.  46  have  been  standard- 
ized by  the  Society  of  Automobile  Engineers.  Each  of  these  types 
is  made  in  three  different  sizes,  A,  B  and  C,  and  the  following 
formulas  give  the  dimensions  of  the  various  parts  of  the  bore, 
while  the  corresponding  parts  of  the  shaft  are  made  one-thou- 
sandth of  an  inch  less  on  the  smaller  shaft  diameters  and  two 
one-thousandths  on  the  larger  sizes. 

For  the  six-spline  type  shown  in  Fig.  46 (a),  the  formula  for  the 
width  b  of  the  spline  is  the  same  for  all  three  sizes;  the  other  di- 
mensions, however,  vary. 


FIG.  46. 


For  6-A,  d  =  0.90  D 

b  =  0.25  D 
t  =  0.05  D 
For  6-5,  d  =  0.85  D  (108) 

t  =  0.075  D 

For  6-C,  d  =  0.80  D 

t  =  0.10  D 

As  in  the  case  of  the  six  splines,  the  width  b  for  the  three  sizes 
of  the  ten-spline  fitting  shown  in  Fig.  46(6)  is  kept  constant. 
The  various  proportions  are  given  by  the  following  formulas: 


(109) 


For  10-A, 

d  =  0.91  D 

b  =  0.156  D 

t  =  0.045  D 

For  10-B, 

d  =  0.86  D 

t  =  0.07  D 

For  10-C, 

d  =  0.81  D 

t  =  0.095  D 

122 


COTTER  JOINTS 


[CHAP.  V 


COTTER  JOINTS 

A  cotter  is  a  cross-key  used  for  joining  rods  and  hubs  that  are 
subjected  to  a  tension  or  compression  in  the  direction  of  their 
axis,  as  in  a  piston  rod  and  its  cross-head  ;  valve  rod  and  its  stem  ; 
a  strap  end  and  its  connecting  rod. 

98.  Analysis  of  a  Cotter  Joint.  —  In  Fig.  47  is  shown  one  method 
of  joining  two  rods  through  the  medium  of  a  cotter,  the  rod  being 
loaded  axially  as  shown.  The  joint  may  fail  in  any  one  of  the  ten 
ways  discussed  below. 

(a)  Rods  may  fail  in  tension.  —  The  relation  between  the 
external  force  P  and  the  internal  resistance  of  the  rod  is  given  by 
the  following  formula: 


p  = 


( 


FIG.  47. 


(6)  Failure  of  the  rod  across  the  slot. — Equating  the  external 
force  to  the  tension  in  the  rod  across  the  slot,  we  get 


(in) 


(c)  Failure  of  the  socket  across  the  slot.  —  Equating  the  external 
force  to  the  internal  resistance  due  to  the  tension  in  the  socket 
across  the  slot,  we  find  that 


P  = 


-  d\)-(D  - 


(112) 


(d)  Cotter  may  shear.  —  Due  to  the  force  P,  the  cotter  may 
fail  by  double  shearing;  hence  the  relation  between  the  load  and 
stress  is  as  follows: 

P=.2btS,  (113) 

(e)  Rod  end  may  shear.  —  To  prevent  the  rod  end  from  failing 


ART.  98]  COTTER  JOINTS  123 

due  to  double  shearing  through  the  length  a,  the  following  ex- 
pression may  be  used  to  determine  the  minimum  value  of  a : 

P  =  2adlS8  (114) 

(/)  Socket  end  may  shear. — The  dimension  c  must  be  made 
long  enough  so  that  the  end  of  the  socket  will  not  fail  by  double 
shearing.  Equating  the  internal  resistance  to  the  force  P,  we 
get 

P  =  2c(D-  d,)&  (115) 

(g)  Socket  or  cotter  may  crush. — The  external  force  may  crush 
either  the  cotter  or  the  socket  along  the  surfaces  AB  and  CE; 
hence,  liberal  surfaces  must  be  provided.  The  following  expres- 
sion gives  the  relation  between  the  load  and  stresses: 

P  =  t(D-dl)Sb  (116) 

(h)  Rod  or  cotter  may  crush. — To  prevent  the  rod  or  cotter 
from  crushing  along  the  surface  FG,  the  relation  expressed  by  the 
following  formula  must  be  fulfilled: 

P  =  tdiSb  (117) 

The  cotter  joint  illustrated  by  Fig.  47  may  also  be  used  for  a 
class  of  service  in  which  the  force  P  may  be  reversed  in  direction, 
thus  producing  a  compression  in  the  rod  in  place  of  a  tension. 
Such  a  loading  will  then  call  for  an  investigation  of  the  collar. 

(i)  Collar  may  shear  off. — Due  to  the  compression  in  the 
rods,  the  collar  may  shear  off;  whence 

|P  =  ird.eS,  (118) 

(j)  Collar  may  crush. — To  prevent  crushing  of  the  collar,  the 
surface  in  contact  must  be  made  large  enough  so  that  the  follow- 
ing relation  between  the  load  and  stress  is  satisfied: 

P  =  I  (dl  -  dl)Sb  (119) 

The  taper  on  the  cotter  should  not  be  made  excessive,  or 
trouble  may  be  experienced  due  to  the  loosening  of  the  cotter 
when  the  joint  is  under  load.  To  prevent  such  loosening,  the 
cotter  may  be  provided  with  a  set  screw.  A  practical  taper  is 
J£  inch  per  foot,  but  this  may  be  increased  to  1J£  inches  per 
foot,  provided  some  locking  device  is  applied  to  the  cotter.  The 
cotter  instead  of  being  made  square-ended  as  shown  in  Fig.  47, 


124 


TAPER  PINS 


[CHAP.  V 


is  more  often  made  with  semi-circular  edges.     This  method  of 
making  the  cotter  possesses  the  following  advantages: 

1.  Sharp  corners  that  are  liable  to  start  cracks  are  avoided. 

2.  The  shearing  area  at  the  sides  of  the  slots  is  increased  con- 
siderably. 

3.  The  slots  with  semi-circular  ends  cost  less  to  make. 

PIN  JOINTS 

In  Art.  90(/i),  the  use  of  round  and  taper  pins  in  the  form  of 
keys  was  discussed,  and  in  the  following  articles  additional  uses 
of  pins  will  be  taken  up.  These  uses  are  as  follows: 

(a)  For  rigid  fastenings  in  which  the  pins  are  so  placed,  that 
they  are  either  in  single  or  double  shear  due  to  the  external  force. 

(6)  For  joining  two  rods  which  require  a  certain  amount  of 
motion  at  the  joint. 

99.  Taper  Pins. — Taper  pins  properly  fitted  form  a  cheap  and 
convenient  means  of  fastening  light  gears,  hand  wheels  and  levers 


(b) 


FIG.  48. 


to  shafts  that  transmit  a  small  amount  of  power.  They  may 
also  be  used  for  making  a  connection  between  two  rods,  similar 
to  the  cotter  joint  described  in  Art.  98.  The  common  method  of 
applying  taper  pins  is  illustrated  in  Fig.  48 (a) ;  but  this  method  is 
applicable  to  the  transmission  of  a  torque  in  only  one  direction. 
If  the  machine  parts  are  subjected  to  alternating  stresses,  as 
would  be  the  case  in  a  coupling  between  the  valve  rod  and  the 
valve  stem,  the  taper  pins  should  be  given  a  slight  clearance 
similar  to  that  provided  for  the  cotter  in  Fig.  47. 

Another  very  important  application  of  taper  pins  is  their  use 
as  dowel  pins  on  bearing  flanges,  and  all  forms  of  brackets  and 
attachments  on  machine  frames.  The  main  function  of  dowel 


ART.  100]  TAPER  PINS  125 

pins  is  to  form  a  convenient  means  of  locating  accurately  a 
bearing  or  bracket,  since  cap  screws  and  studs  cannot  be  relied  on 
for  that  purpose.  If  the  taper  pins  are  fitted  correctly  and 
located  properly,  no  trouble  is  experienced  in  reassembling  the 
machine  parts  after  being  dismantled  for  repair  or  other  purposes. 
In  Fig.  36  is  shown  the  application  of  two  taper  dowel  pins  c  on 
the  flange  of  a  sond  bearing.  It  should  be  observed  that  these 
pins  are  not  diametrically  opposite,  though  in  this  case  they 
could  have  been  located  symmetrically,  since  the  location  of  the 
oil  hole  in  the  bearing  would  insure  the  correct  assembling. 
However,  many  symmetrical  castings  or  brackets  are  used,  and 
the  location  of  the  dowel  pins  as  illustrated  in  Fig.  36  may  obviate 
a  lot  of  unnecessary  work.  Another  function  of  dowel  pins, 
which  in  many  cases  is  of  great  importance,  is  to  make  these  pins 
take  the  shearing  action  due  to  the  external  load,  thus  relieving 
the  cap  screws  or  studs  from  such  action. 

As  mentioned  in  Art.  90  (h),  standard  taper  pins  cost  but 
little,  and  the  various  sizes  and  lengths  listed  in  Table  28  are 
carried  regularly  in  stock  by  the  various  manufacturers.  The 
taper  adopted  by  the  manufacturers  is  one-fourth  inch  per  foot. 
These  standard  taper  pins  have  no  provision  on  the  head  or  point 
that  will  allow  for  upsetting  the  ends,  if  desired.  Provisions 
for  upsetting  can  be  made  by  having  the  heads  and  points  tapered, 
which  would  also  facilitate  the  driving  of  the  pin  into  the  machine 
part  as  well  as  its  removal.  For  removing  large  dowel  pins  such 
as  are  used  in  locating  the  housings  on  planers  and  heavy  milling 
machines,  the  taper  pin  is  provided  at  the  large  end  with  a 
threaded  shank  which  is  fitted  with  a  nut;  hence  to  remove  the 
dowel  pin  merely  back  out  the  pin  by  screwing  up  the  nut. 

Occasionally  a  threaded  shank  is  provided  at  the  small  end  of 
the  pin,  which  if  fitted  with  a  nut  forms  an  effective  means  of 
retaining  a  pin  having  a  steep  taper.  When  the  taper  pin  is 
used  as  a  fastening  similar  to  that  shown  in  Fig.  48 (a),  the  large 
diameter  D  of  the  pin  is  made  from  one-fourth  to  one-third  of  the 
diameter  of  the  rod  or  shaft  through  which  the  pin  passes.  The 
length  L  Fig.  48(6)  is  chosen  so  that  the  pin  projects  a  small 
amount  on  each  side  of  the  hub,  though  not  enough  to  make  it 
dangerous.  Table  28  also  contains  information  pertaining  to  the 
standard  reamers  that  are  used  with  the  standard  taper  pins. 

100.  Rod  and  Yoke  Ends. — Various  forms  of  pin  joints  are 
used  for  connecting  together  two  or  more  rods  and  at  the  same 


126 


ROD  AND  YOKE  ENDS 


[CHAP.  V 


•  4- 

i 

•• 

} 

r 

-3- 
-« 

..«!.                   ^ 
•*J 

X 

v_ 

J, 

e 

0 

V 

«A 

> 

>p 

|! 

|i 

1 

LJ 


r3- 


r 


r4i 


(c) 


(a) 


(b) 
FIG.  49 


-1 


-51 


Fio.  60. 


ART.  100] 


ROD  AND  YOKE  ENDS 


127 


time  permitting  a  certain  amount 
of  motion  at  the  joint.  Such  joints 
are  called  rod  and  yoke  ends  or 
knuckle  joints  and  are  used  in  prac- 
tically all  classes  of  machinery.  In 
Fig.  49  are  shown  the  standard 
drop-forged  rod  and  yoke  ends 
adopted  by  the  Society  of  Auto- 
mobile Engineers,  and  the  propor- 
tions thereof  are  included  in  Table 

30.  It  should  be  noticed  that  the 
yoke  ends  are  made  in  two  types, 
namely,    the    adjustable   and    the 
plain,  illustrated  by  Fig.  49 (a)  and 
(&),  respectively. 

The  sizes  of  yoke  and  rod  ends 
used  in  the  automobile  industry  do 
not  cover  a  wide  range,  and  in 
order  to  meet  the  demand  for  yoke 
and  rod  ends  adapted  to  general 
use,  several  manufacturers  of  drop 
forgings  carry  such  parts  in  stock. 
In  Fig.  50  are  shown  finished  plain 
rod  and  yoke  ends  that  are  a 
standard  product  of  The  Billings 
and  Spencer  Co.  of  Hartford, 
Conn.  The  dimensions  indicated 
in  Fig.  50  are  included  in  Table 

31.  The    plain    shanks    of   these 
forgings    are    made    of    sufficient 
length  to  permit  welding  them  on 
to  rods  of  any  desired  length. 

The  type  of  rod  end  just  dis- 
cussed has  no  provision  whatever 
for  taking  up  wear  at  the  joint,  and 
in  the  class  of  service  for  which 
they  are  intended,  it  is  not  custo- 
mary to  make  such  provision. 
There  are,  however,  many  places 
where  the  wear  on  the  pin  or  its 
bearing  must  be  taken  up  and 


<* 

WWW 

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per  inch 

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128 


ROD  AND  YOKE  ENDS 


[CHAP.  V 


hence  the  design  of  such  rod  ends  requires  some  knowledge  of 
bearing  and  journal  design.  Such  machine  parts  are  discussed 
in  detail  in  Chapter  XIX. 

TABLE  31. — B.  &  S.  DROP-FORGED  ROD  AND  YOKE  ENDS 


Rod  end 


Yoke  end 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 


K 
KG 

H 
KG 

KG 


1 

IK 
IK 


3 


4iM. 

SMe 


6^2 

7^2 

7K6 


1 

IK 

IK 
IK 


2Me 

2% 


KG 
KG 
KG 


G 


XK6 

IMe 


IMe 


XlG 

KG 


7 
7% 


10 


2K6 
2*Me 


KG 
KG 

KG 
Me 


1 

IK 

IK 


References 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 
Mechanical  Engineers'  Handbook,  by  L.  S.  MARKS,  Editor  in  Chief. 
KENT'S  Mechanical  Engineers'  Pocket  Book. 


CHAPTER  VI 

CYLINDERS,  PLATES  AND  SPRINGS 
CYLINDERS 

In  the  following  discussion,  cylinders  will  be  divided  into  two 
general  classes,  as  follows: 

(a)  Those  having  thin  walls,  as  for  example,  steam  and  water 
pipes,  boiler  shells  and  drums. 

(b)  Those  having  relatively  thick  walls. 

101.  Thin  Cylinders. — In  analyzing  the  stresses  induced  in  the 
walls  of  thin  cylinders  by  an  internal  pressure,  we  shall  assume, 
first,  that  the  stresses  are  distributed  uniformly  over  the  cross- 
section  of  the  cylinder;  and  second,  that  the  restraining  action  of 
the  heads  at  the  ends  of  the  cylinder  is  zero.  Considering  a 
cylinder  having  its  ends  closed  by  heads,  the  internal  pressure 
against  these  heads  produces  a  longitudinal  stress  in  the  walls; 
the  magnitude  of  which  is 

S,  =  £7  (120) 

in  which  d  represents  the  inner  diameter  of  the  cylinder,  p  the 
unit  internal  pressure  and  t  the  thickness  of  the  cylinder  walls. 

Assuming  that  the  above  cylinder  is  cut  by  a  plane  through  its 
axis,  the  resultant  internal  pressure  on  a  section  of  either  half 
cylinder  having  a  length  L  as  pdL]  hence,  the  magnitude  of  the 
tangential  or  hoop  stress  is 

S't  -  &  (121) 

Comparing  (120)  and  (121),  it  is  apparent  that  the  longitudinal 
stress  St  is  one-half  of  the  tangential  stress;  however,  the  true 
tangential  stress  is  even  less  than  that  given  by  (121).  Assum- 
ing that  Poisson's  ratio  has  a  value  of  0.3,  the  effective  tangential 
stress  is 

„        pd       0.3  pd       0.425  pd  . 

St  ="  2t"  ~TT~     ~~r 

Designers  never  use  formula  (122);  they  prefer  (121)  since  the 

129 


130 


THICK  CYLINDERS 


[CHAP.  VI 


thickness  of  the  walls  obtained  by  the  latter,  for  any  assumed  set 
of  conditions,  is  always  greater. 

102.  Thick  Cylinders. — In  a  cylinder  having  walls  that  are 
thick  when  compared  to  the  internal  diameter,  the  stresses  in- 
duced by  an  internal  pressure  p  cannot  be  considered  uniformly 
distributed  as  in  the  preceding  case.  The  tangential  stress,  or 
hoop  tension  as  it  is  frequently  called,  varies  along  the  wall 
thickness,  having  its  greatest  magnitude  at  the  interior  of  the 
cylinder  and  its  minimum  at  the  exterior  surface.  Several 
investigators  have  proposed  formulas  that  are  applicable  to  the 

design  of  thick  cylinders,  among 
the  most  prominent  of  these  be- 
ing Lame,  Clavarino,  and  Birnie. 
(a)  Lame's  formula. — In  the 
Ca.oe  of  a  cylinder  subjected  to 
both  internal  and  external  pres- 
sure, as  shown  in  Fig.  51,  the 
tangential  and  radial  stresses  at 
the  variable  radius  r  are  given, 
according  to  Lame,  by  the  follow- 
ing expressions: 

M  +  ^  (123) 


FIG.  51. 


in  which 
and 


„        pd*  -  p,Z)' 
M~     D*-& 


N 


d2D2  rp  — 


[P  -  Pol 
ID2  -  d2] 


(124) 
(125) 
(126) 


In  order  to  derive  a  formula  that  is  applicable  to  thick  cylinders 
subjected  only  to  internal  pressure,  we  make  po  =  0  in  (125)  and 
(126);  then  the  maximum  tangential  stress  occurs  on  the  inner 
surface  of  the  cylinder,  and  its  magnitude  is 


(127) 


This  is  one  of  the  forms  in  which  the  Lame*  formula  may  be 
used,  but  very  often  it  is  found  that  another  form  is  more  con- 


ART.  102]  BIRNIE'S  FORMULA  131 

venient.     This  may  be  derived  by  clearing  (127)  of  fractions  and 
substituting  (2  1  +  d)  for  D,  whence 

(128) 

(&)  Clavarino*  s  andBirnie's  formulas.  —  In  the  preceding  dis- 
cussion, Poisson's  ratio  of  lateral  contraction  was  not  intro- 
duced, and  for  that  reason  (127)  and  (128)  are  only  approximate. 

According  to  the  maximum-strain  theory  proposed  by  Saint 
Venant,  the  effective  tangential  and  radial  stresses  are  as  follows  : 

St  =  Edt 


in  which  dt  and  8r  represent  the  unit  tangential  and  radial  strains. 
It  is  evident  that  these  strains  or  deformations  depend  on  the 
longitudinal  stress  in  the  walls  of  the  cylinder.  Two  cases  may 
occur,  namely,  a  cylinder  may  have  its  ends  open  or  the  ends  may 
be  closed. 

1.  Cylinder  with  open  ends.  —  In  a  cylinder  having  open  ends, 
the  longitudinal  stress  is  zero;  and  assuming  the  cylinder  to  be 
under  an  internal  pressure,  the  maximum  tangential  stress  is 

St  =  (1  -  ro)  M  +  (1  +  m)  —'  |(130) 

in  which  m  represents  Poisson's  ratio. 

Substituting  the  values  of  M  and  N  from  (125)  and  (126)  in 
(130),  we  get  finally 


Substituting  in  (131),  the  value  of  D  in  terms  of  d  and  t,  we 
have 


'-I 


WS,+  (l-m)p  _  ,1 
S,  -  (1  +  «)  p 


This  formula  is  that  due  to  Birnie  and  applies  only  to  cylinders 
having  open  ends. 

2.  Cylinder  with  closed  ends. — The  second  case  mentioned 
above  is  the  one  of  most  frequent  occurrence,  namely,  that  in 
which  the  ends  of  the  cylinder  under  internal  pressure  are  closed. 
For  this  condition,  the  magnitude  of  the  maximum  effective 


132  CLAVARINO'S  FORMULA  [CHAP.  VI 

tangential  stress  is  given  by  the  expression 

St  =  (1  -  2m)  M+(l  +  m)ji*  (133) 

from  which 

St  =  D2P_  d2  [(1  -  2  w)  d2  +  (1  +  m)  D2]  (134) 

If  an  expression  for  t  is  desired,  it  may  be  obtained  from  (134) 
by  substituting  for  D  its  value  in  terms  of  t  and  d'}  whence 


d      IS,+  (l-2m)p 
-- 


This  expression  is  known  as  Clavarino's  formula  and  applies 
to  all  cylinders,  under  internal  pressure,  having  closed  ends. 

For  values  of  ra  to  be  used  in  the  above  formulas,  refer  to 
Table  1. 

PLATES 

The  various  formulas  in  common  use  for  determining  the 
strength  of  flat  plates  subjected  to  various  methods  of  loading 
are  generally  based  upon  some  arbitrary  assumption  regarding 
the  critical  section  or  the  reactions  of  the  supports.  Grashof, 
Bach,  Merriman,  and  others  have  treated  this  subject  from  a 
mathematical  standpoint,  and  the  Various  formulas  proposed  by 
these  investigators  give  results  that  agree  fairly  well  with  the 
experimental  results  obtained  by  Bach,  Benjamin,  Bryson,  and 
others.  Flat  plates  subjected  to  various  methods  of  loading  are 
of  frequent  occurrence  in  machines,  and  the  formulas  in  the 
following  articles  are  those  proposed  by  Prof.  Bach.  They 
are  reliable  and  are  comparatively  easy  to  apply  to  any  given 
set  of  conditions.  It  should  be  understood,  however,  that  these 
formulas  apply  only  to  the  plain  flat  plate  and  not  to  plates  hav- 
ing a  series  of  reenforcing  ribs  that  are  commonly  used  when  the 
plates  are  cast. 

103.  Rectangular  Plates.  —  In  arriving  at  formulas  for  the 
strength  of  rectangular  plates,  the  critical  section  is  taken  as  pass- 
ing through  the  center  of  the  plate,  and  the  part  to  one  side  of 
this  section  is  treated  as  a  cantilever  beam.  The  location  of  the 
critical  section  is  determined  by  experiments,  and  for  rectangular 
plates  made  of  homogeneous  material,  Bach  found  that  failure 
does  not  always  occur  along  a  diagonal  as  in  the  case  with  square 


ART.  104] 


RECTANGULAR  PLATES 


133 


plates.  However,  in  establishing  a  general  formula,  it  is  usually 
assumed  that  the  line  of  maximum  stress  lies  along  the  diagonal. 
(a)  Uniformly  loaded. — Consider  a  rectangular  plate  of 
thickness  t,  of  length  a  and  of  breadth  6,  as  supported  at  the 
periphery  and  subjected  to  a  pressure  p  that  is  uniformly  dis- 
tributed; then,  according  to  Bach, 


t  =  b 


(136) 


in  which  K  is  a  coefficient  depending  upon  the  method  of  sup- 
porting the  periphery  of  the  plate,  the  condition  of  the  surface  of 
the  plate,  the  initial  force  required  to  make  a  tight  joint,  and  the 
material  used  for  making  the  tight  joint.  The  values  of  K  for 
cast  iron  and  mild  steel  for  various  conditions  of  supporting  the 
loaded  plate  are  given  in  Table  32. 

TABLE  32. — VALUES  OF  COEFFICIENTS  K,  Ki,  K2  AND  K3 


Material 

Condition  of 
support 

K 

Ki 

Kz 

Ki 

f 

Free 

0  565 

2  6-3  0 

0  282 

1  3-1  5 

Cast  iron  j 

Fixed  

0.375 

0.187 

Free 

0  360 

0  180 

Mild  steel  { 

Fixed 

0  240 

0  120 

(b)  Central  loading. — Suppose  now  that  a  rectangular  plate 
having  the  same  dimensions  as  the  one  discussed  previously  be 
supported  at  the  periphery  and  loaded  centrally  by  a  load  Q ;  then 
the  thickness  may  be  determined  by  the  following  expression: 


t  = 


abQK, 


(137) 


For  a  cast-iron  plate  supported  freely,  the  value  of  KI  as  deter- 
mined experimentally  by  Bach  varies  from  2.6  to  3.0. 

104.  Square  Plates.  —  For  similar  conditions  of  loading  and 
supporting  the  plate,  the  formulas  for  the  thickness  of  square 
plates  may  be  derived  directly  from  the  corresponding  formula 
pertaining  to  rectangular  plates.  Therefore,  for  uniformly  dis- 
tributed pressure,  the  thickness  is 


(138) 


134 


CIRCULAR  PLATES 


[CHAP.  VI 


For  a  square  plate  centrally  loaded,  it  is 


(139) 


For  values  of  K2  and  KS}  consult  Table  32. 

105.  Circular  Plates.— (a)  Pressure  uniformly  distributed. — The 
thickness  of  a  circular  plate  having  a  diameter  a,  and  which  is 
supported  around  its  circumference  and  subjected  to  a  uniformly 
distributed  pressure,  is  determined  by  the  following  formula: 


(140) 


and  its  deflection  is  given  by  the  expression 


A  =  ^  (141) 

In  (140)  and  (141),  K±  and  Kb  represent  coefficients  which 
depend  upon  the  method  of  support  as  well  as  the  method  and 
materials  used  in  making  the  joint  tight.  Values  of  these 
coefficients  are  given  in  Table  33. 

TABLE  33. — VALUES  OF  COEFFICIENTS  K4,  K5,  K6  AND  K? 


Material 

Condition  of 
support 

Kt 

K6 

Kt 

Ki 

Cast  irorH 

Free  
Fixed  

0.30 
0.20 

0.038 
0.010 

1.43 

0.1-0.125 

Free  

0  19 

Mild  steely 

1 

Fixed  

0.13 

(6)  Central  loading. — For  a  flat  circular  plate  supported  freely 
around  the  circumference  and  subjected  to  a  load  Q  at  the  center 

which  is  considered  as  distributed  uniformly  over  the  area  -j-> 
the  thickness  is  given  by  the  following  formula : 

t  =  A/tfJl-^l^-  (142) 


K  \^      2d 

=  V^6  L1  "  3 


The  deflection  caused  by  the  load  Q  may  be  determined  by  the 
relation 


A  = 


Et* 


For  values  of  the  coefficients  KG  and 


(143) 
consult  Table  33. 


ART.  106] 


CYLINDER  HEADS 


135 


According  to  Grashof,  the  thickness  of  a  circular  plate  fixed 
rigidly  around  the  circumference  and  loaded  centrally  by  the 
load  Q  may  be  calculated  by  the  relation 


t  = 


.435  Q 


a 
;ed 


(144) 


If  the  deflection  is  desired,  the  following  expression  may  be  used : 

0.055  Pa2 


A  = 


Et* 


(145) 


106.  Flat  Heads  of  Cylinders.  —  (a)  Cast  heads.  —  In  the  case 
of  a  cast-iron  cylinder  having  the  flat  head  cast  integral  with  the 
sides,  as  shown  in  Fig.  52,  the  allowable  stress  in  the  head, 
according  to  Bach,  is  given  by  the  relation 


S  =  0.8 


(146) 


1 

-s 

\ 

FY//////, 

^%^? 

f    » 

m  y\ 

p 

(a) 


Cb) 


FIG.  52. 


(6)  Riveted  heads. — The  stress  in  the  flat  head  riveted  to  a 
cylindrical  shell,  according  to  Bach,  is 


(147) 


in  which  the  various  symbols  have  the  same  meaning  as  above. 

107.  Elliptical  Plates. — (a)  Pressure  uniformly  distributed. — 
Plates  having  an  elliptical  form  are  frequently  met  with  in  en- 
gineering designs;  for  example,  handhole  plates  and  covers  for 
manholes  in  pressure  vessels.  The  following  formula,  due  to 
Bach,  gives  the  thickness  of  an  elliptical  plate  subjected  to  a 


136 


HELICAL  SPRINGS 


[CHAP.  VI 


uniformly  distributed  pressure,  and  whose  major  and    minor 
axes  are  a  and  b,  respectively: 


t  =  K8b 


P 


1+  - 


(148) 


The  values  of  KB  for  cast  iron  and  mild  steel,  and  for  two  condi- 
tions  of  supporting 
TABLE  34.-VALuES  OF  COEFFICIENTS  K8  AND  K9  the  plate>  afe  giyen 

""  in  Table  34. 

(b)  Central  load- 
ing.— The  thickness 
of  an  elliptical  plate 
supported  around 
the  periphery  and 
subjected  to  a  load 
Q  Sit  the  center  is  given  by  the  following  expression: 


Material 

Condition  of 
support 

K, 

K9 

Cast  iron.  | 

Free  
Fixed.  .  . 

0.82 
0  58 

0.85 
0  77 

Free 

0  60 

Mild  steel  I 

Fixed  

0.46 

::::::: 

in  which  c  represents  the  ratio  of  the  minor  axis  b  to  the  major 
axis  a.  For  values  of  Kg,  for  various  conditions  of  loading,  con- 
sult Table  34. 

SPRINGS 

Springs  are  made  in  a  variety  of  forms,  depending  upon  the 
class  of  service  for  which  they  are  intended.  Among  the  com- 
mon forms  used  to  a  considerable  extent  in  connection  with 
machinery,  are  the  following:  (a)  Helical  springs;  (6)  spiral 
springs;  (c)  conical  springs;  (d)  leaf  springs. 

108.  Helical  Springs. — Helical  springs  are  used  chiefly  to  resist 
any  force  or  action  that  tends  to  lengthen,  shorten,  or  twist  them. 
The  wire  or  bar  used  to  make  this  type  of  spring  may  have  a  cir- 
cular, square,  or  rectangular  cross-section.  The  stresses  induced 
in  the  material  of  a  helical  spring  subjected  to  an  extension  or  a 
compression  consist  of  a  tension  combined  with  secondary  stresses, 
such  as  tensile  and  compressive  due  to  a  bending  action.  The 
latter  stresses  are  generally  not  considered  in  the  development  of 
suitable  formulas  for  the  permissible  load  and  the  deflection. 

(a)  Circular  wire. — The  method  of  procedure  in  arriving  at 


ART.  108]  HELICAL  SPRINGS  137 

the  relations  existing  between  the  axial  deflection  and  the  axial 
load  for  a  helical  spring  made  of  round  wire  is  as  follows : 

Let  D  =  mean  diameter  of  the  coils. 
E9  =  torsional  modulus  of  elasticity. 
Q  =  axial  load  on  the  spring. 
d  =  diameter  of  the  wire. 
n  =  number  of  coils. 
p  =  pitch  of  coils. 
A  =  total  axial  deflection. 

The  stresses  at  any  section  of  the  bar  at  right  angles  to  the  axis 
of  the  spring  are  those  due  to  the  torsional  moment  —~-  and  to  the 

bending  action,  the  effect  of  the  latter  being  disregarded.     Apply- 
ing the  formula  for  torsional  stress  from  Art.  10,  we  have 

(150) 

In  determining  the  safe  stress  for  any  given  case  by  means  of 
(150),  the  magnitude  of  Q  must  be  taken  as  the  greatest  load  that 
will  ever  come  upon  the  spring.  Frequently  (150)  is  used  for 
calculating  the  safe  load  that  a  spring  will  carry,  or  it  may  be 
used  for  arriving  at  the  size  of  the  wire  required  for  a  given 
load,  safe  working  stress,  and  diameter  of  coil. 

The  total  length  of  the  bar  required  to  make  the  spring  is 
^A/TrD2  +  p2  or  approximately  irnD,  and  according  to  Art.  10, 
the  angular  deflection  of  a  bar  having  the  length  just  given,  is 

g  _  360 inDS, 
aEi8 

The  axial  deflection  of  the  spring  is  evidently  given  by  the 
following  formula: 

A  =  'r?gp  (152) 

Substituting  the  value  of  S,  from  (150)  in  (152) 


Having  given  the  load  Q  and  the  corresponding  deflection 
A,  (153)  will  be  found  useful  for  determining  the  required  number 
of  coils  n,  by  assuming  values  for  the  size  of  wire  and  the  diameter 


138 


HELICAL  SPRINGS 


[CHAP.  VI 


of  the  coils.     The  formula  for  the  deflection  as  given  by  (152) 
may  be  used  for  calculating  the  safe  deflection. 

In  designing  helical  springs,  the  following  method  of  procedure 
is  suggested : 

1.  By  means  of  (150),  determine  the  diameter  of  the  coil  re- 
quired for  the  given  load  and  assumed  values  of  the  fiber  stress 
and  size  of  wire.     The  results  obtained  may  have  to  be  rounded 
out  so  as  not  to  get  an  odd  size  of  arbor  upon  which  the  spring 
is  made. 

2.  Having  arrived  at  a  proper  dimension  for  the  diameter  of 
the  coil,  the  deflection  may  be  determined  by  means  of  (153), 
provided  we  know  the  number  of  coils  required,  or  if  the  deflec- 
tion is  fixed  by  the  surrounding  conditions,  the  number  of  coils 
required  may  be  calculated  by  means  of  (153). 

(b)  Bar  having  rectangular  cross-section. — For  helical  springs 
made  of  a  wire  or  bar  having  a  rectangular  cross-section  b  X  h, 
as  shown  in  Fig.  53,  the  relation  between  the  fiber  stress  S,  and 


FIG.  53. 


the  external  load  Q  is  obtained  by  equating  the  external  moment 
to  the  moment  of  resistance;  whence 


S,  = 


9QD 

4b2h 


(154) 


This  formula  is  used  to  establish  the  size  of  the  wire  for  any 
given  load  and  safe  stress,  or  it  may  be  used  to  check  the  stress 
having  given  the  load  and  size  of  wire. 

According  to  the  Mechanical  Engineers'  Handbook,  the  axial 
deflection  of  the  spring  may  be  calculated  by  the  following 

formula  : 

2.83  nQD*  (b2  +  V) 


WE. 


, 


If  an  expression  for  the  axial  deflection  is  desired  in  terms  of  the 


ART.  109]  HELICAL  SPRINGS  139 

safe  stress  Sa,  the  value  of  Q  obtained  from  (154)  is  substituted 
in  (155);  whence 


A-      . 

The  method  of  procedure  to  be  used  in  the  design  of  a  helical 
spring  constructed  of  a  rectangular  bar,  as  shown  in  Fig.  53, 
is  the  same  as  that  suggested  in  (a)  above. 

(c)  Bar  having  square  cross-section. — In  many  installations 
requiring  helical  springs,  square  wire  is  preferred  to  the  rectangu- 
lar. By  making  b  =  h  in  (154),  (155)  and  (156),  we  obtain  the 
desired  equations  necessary  for  designing  springs  constructed  of 
square  wires. 

For  a  given  load  and  assumed  fiber  stress,  the  size  of  the  wire 
or  bar  may  be  calculated  by  means  of  the  following  formula: 


The  axial  deflection  may  be  determined  from 

5.65  nQD3 

or  from 

A  =  *«ypk  (159) 

For  the  method  of  procedure,  the  suggestions  given  in  (a) 
above  may  be  followed. 

109.  Concentric  Helical  Springs. — The  springs  used  in  many 
automobile  clutches,  as  well  as  those  used  on  railway  trucks, 
consist  of  two  concentric  helical  coils,  both  of  which  are  neces- 
sarily deflected  equal  amounts,  since  their  free  and  solid  lengths 
are  made  equal.  The  springs  used  on  railway  trucks  are  gener- 
ally made  of  round  bars,  while  those  used  for  automobile  clutches 
are  made  of  round,  rectangular  and  square  stock.  In  actual 
construction,  the  adjacent  coils  of  concentric  springs  are  wound 
right  and  left  hand  so  as  to  prevent  any  tendency  to  bind.  In 
the  design  of  concentric  springs  in  which  the  same  grade  of 
material  is  employed,  an  attempt  should  be  made  to  get  ap- 
proximately the  same  stresses  in  the  various  coils.  With  the 
use  of  round  wire,  the  latter  condition  is  met  by  making  the  ratio 

-T  the  same  for  all  coils,  as  the  following  analysis  shows : 


140  HELICAL  SPRINGS  [CHAP.  VI 

Using  the  same  notation  as  before  and  representing  the  solid 
length  of  the  spring  by  H,  but  adopting  the  subscripts  2  and  1 
to  the  various  d  mensions  of  the  inner  and  outer  coils  respectively, 
it  follows  from  (152)  that  the  stress  in  the  material  of  the  inner 
coils  of  a  double  helical  spring  is 


s  ..      i  /\ 

7T#2       \Dj 

and  that  in  the  outer  coils 

Ai#,   M\2 

s' 


Now,  assuming  that  the  deflections  and  the  solid  heights  are  to 
be  the  same  for  the  two  coils,  it  is  evident  that  for  equal  stresses 

Di_D* 

d,       d,' 

Since  the  ratio  -j  is  the  same  for  both  coils,  it  follows  that  the 

lengths  of  the  bars  from  which  the  separate  coils  are  made  will 
be  the  same. 

110.  Helical  Springs  for  Torsion.  —  Helical  springs  are  also 
used  to  resist  a  torsional  moment  T  by  having  one  end  held 
rigidly  while  the  other  is  relatively  free.  Such  springs  are  in- 
variably made  from  bars  having  a  rectangular  or  square  cross- 
section.  The  material  of  the  spring  is  subjected  to  a  bending 
stress  having  a  magnitude  as  follows: 

S  =  g,  (162) 

in  which  h  is  the  width  and  b  the  radial  thickness  of  the  spring 
stock. 

The  linear  deflection  according  to  the  Mechanical  Engineers' 
Handbook  is 


in  which  the  total  length  L  of  the  bar  may  be  assumed  equal  to 
irnD,  as  in  Art.  108(a). 

For  springs  made  of  square  wire,  the  formulas  for  stress  and 
deflection  may  be  derived  from  (162)  and  (163)  by  making  h  =  b. 

111.  Spiral  Springs. — The  spiral  spring  is  used  but  little  in 
machine  construction,  and  then  only  for  light  loads.     It  consti- 


ART.  112] 


CONICAL  SPRINGS 


141 


tutes  what  is  commonly  called  a  torsional  spring  and  the  material 
used  in  its  construction  is  subjected  to  a  bending  stress.  Letting 
h  represent  the  width  and  b  the  radial  thickness  of  the  spring 
material,  the  moment  of  the  external  force  Q  must  equal  the 
internal  resistance;  hence 


S 


3QD 

hb2 


(164) 


The  following  expression  for  the  linear  deflection  A  of  a  spiral 
spring  is  that  given  in  the  Mechanical  Engineer's  Handbook. 

QLD*      LSD 

A     —    _ —   . 

A  "  4  El         bE 


(165) 


in  which  L  represents  the  length  of  the  straightened  spring  and 

the  other  symbols  are  as  in  the  preceding 

articles. 

112.  Conical  Springs. — Conical  springs 
are  generally  used  to  resist  a  compression 
and  are  made  of  round  or  rectangular  stock. 
They  are  applicable  where  the  space  is 
limited,  and  where  there  is  no  necessity  for 
great  deflections.  The  following  formulas 
derived  from  the  Mechanical  Engineers' 
Handbook  may  serve  for  determining  the 
proportions  of  such  springs: 

For  a  conical  spring  made  of  round  stock 
and  loaded  as  shown  in  Fig.  54,  the  shear- 
ing-stress in  the  material  is  as  follows: 

SQD2 


(166) 


The  axial  deflection  for  n  turns  or  coils  is  given  by  the  following 
expression  : 


J4  El 


(Dl  +  DlD, 


(167)- 


If  the  expression  for  the  deflection  is  desired  in  terms  of  the 
safe  stress,  we  have 


V  01  +  Dfri  +  DlD2  +  Dl) 


(168) 


A  conical  spring  made  of  rectangular  stock  is  shown  in  Fig.  55. 


142 


LEAF  SPRINGS 


[CHAP.  VI 


The  torsional  stress  in  the  material  of  such  a  spring  may  be 

calculated  by  the  formula 

cmn. 

(169) 


The  axial  deflection  in  terms  of  the  load  Q  is 
0.71  nQ  (b»  +  ft«)  (DJ  +  Dfoi  +  D 


In  terms  of  the  safe  stress,  the  axial  deflection  is 

_  0.315  n  (b2  +  /i2)  (Z>2  - 

~WD2ES 


FIG.  56. 


(170) 


(171) 


113.  Leaf  Springs. — Leaf  springs  are  made  in  various  forms 
some  of  which  are  shown  in  Fig.  56.  The  first  form  shown  is 
called  the  full  elliptic,  the  second  semi-elliptic  and  the  ordinary 


ART.  114]  SEMI-ELLIPTIC  SPRINGS  143 

flat  leaf  spring  is  represented  by  Fig.  56  (c).  In  all  of  the  forms 
shown,  the  various  leaves  are  banded  tightly  together,  and,  as 
usually  constructed,  each  type  has  one  or  more  full-length  leaves, 
sometimes  called  master  leaves,  while  the  remaining  leaves  are 
graduated  as  to  length.  With  this  construction  it  is  evident  that 
the  master  leaves  held  rigidly  by  the  band  constitute  a  cantilever 
beam  of  uniform  cross-section,  while  the  remaining  leaves  form 
approximately  a  cantilever  beam  of  uniform  strength.  From 
the  theory  of  cantilever  beams  we  find  that  the  deflection  of  the 
graduated  leaves  for  the  same  load  and  fiber  stress  will  be  50  per 
cent,  greater  than  that  of  the  master  leaves.  Furthermore, 
when  the  leaves  are  banded  together  without  any  initial  stress, 
the  master  leaves  and  the  graduated  leaves  will  deflect  equal 
amounts,  thus  subjecting  the  former  to  a  higher  fiber  stress.  It 
is  possible  to  make  the  fiber  stresses  in  the  two  parts  of  the  spring 
approximately  equal  by  separating  them  by  a  space  equal  to  the 
difference  between  the  two  deflections  before  putting  the  band 
in  place;  hence,  when  the  band  is  in  place  and  the  spring  is  un- 
loaded an  initial  stress  is  set  up  in  the  leaves.  It  is  customary  to 
consider  one  of  the  master  leaves  as  a  part  of  the  cantilever  beam 
of  uniform  strength. 

114.  Semi-elliptic  Springs. — The  following  analyses  and  for- 
mulas pertaining  to  semi-elliptic  springs  are  due  to  Mr.  E.  R. 
Morrison,  who  probably  was  the  first  to  take  into  account  the 
effect  of  the  initial  stress  due  to  the  band  located  at  the  mid- 
dle of  elliptic  and  semi-elliptic  springs  as  used  in  automobile 
construction. 

Let  Q  =  total  load  on  the  spring. 

Qg  =  load  coming  upon  one  end  of  the  graduated  leaves. 
Qm  =  load  coming  upon  one  end  of  the  master  leaves. 
Sa  =  maximum  fiber  stress  in  the  graduated  leaves. 
Sm  =  maximum  fiber  stress  in  the  master  leaves 
n  =  total  number  of  leaves  in  the  spring. 
ng  =  total  number  of  graduated  leaves. 
nm  =  total  number  of  master  leaves. 

(a)  Initial  space  between  leaves. — From  a  study  of  cantilever 
beams,  it  is  evident  that  in  order  to  satisfy  the  condition  of 
equal  stress  in  the  graduated  and  master  leaves,  the  following 
equation  will  result: 

6LQ,  _6LQ 

~ 


144  SEMI-ELLIPTIC  SPRINGS  [CHAP.  VI 

from  which 

Qo    =    Qrn  (173) 

nff       nm 

The  difference  between  the  deflections  of  the  graduated  and 
master  leaves  is  given  by  the  following  expression: 


Since  —  -  =  75—  >  it  follows  that  the  depth  of  the  space  which 

Tim  ^n 

must  be  provided  between  the  two  parts  of  the  spring  before  they 
are  banded  together  is 


ZbE 

(b)  Pressure  due  to  the  central  band.  —  If  the  total  pressure 
exerted  by  the  central  band  upon  the  leaves  is  Qb,  then  the  deflec- 

tion of  the  graduated  leaves  due  to  -7^,  which  is  the  pressure 
exerted  by  the  band  upon  each  cantilever,  is  as  follows: 

A'  =  ^^  (176) 

Afl       hb*Ena 

The  pressure  -~  also  produces  a  deflection  in  the  master  leaves, 
the  magnitude  of  which  is 

A'   =   *^&  (177) 

"       WEnm 

Combining  (176)  and  (177),  we  have 

(178) 


Since  the  total  deflection  produced  by  the  band  is  equal  to  the 
depth  of  the  space  provided  between  the  two  parts  of  the  spring, 
it  follows  that 


2ng 

m  ~ 


from  which 


Combining  (177)  and  (179),  we  get  the  following  expression  for 
the  magnitude  of  the  pressure  exerted  by  the  band: 

(180) 


ART.  115]  MATERIALS  FOR  SPRINGS  145 

The  expression  for  Qb  just  derived  may  be  simplified  by  letting 
nm  =  kn.  Since  n  =  ng  +  nm,  it  follows  that  ng  =  n(l  —  k). 
Substituting  these  values  of  ng  and  nm  in  (180),  we  get 

k  (1  -  fe)  Q  ,       . 

Qb=    -2T*~ 

(c)  Deflection  of  spring  due  to  Q.  —  The  deflection  A  of  the 
spring  due  to  Q  is  determined  by  taking  the  difference  between 
the  total  deflection  of  the  graduated  leaves  and  that  due  to  the 
band  as  given  by  (178);  whence 

6L'Q,  Znm         L*Q 

0          °       hb*Eng       3  nm  +  2  ng  nhb*E 
or 


Now  since  Q  =  2  (Qg  +  QJ  =     ^  ,  we  get  finally  that  the 
deflection  A  due  to  the  load  Q  is 

-mfia        <•»> 

In  the  above  discussion,  the  effect  of  friction  between  the  leaves 
was  not  considered. 

(d)  Full  elliptic  springs.  —  The  analysis  given  for  the  semi- 
elliptic  springs  also  applies  to  the  full  elliptic  type,  except  that 
the  total  deflection  A  will  be  double  that  of  a  semi-elliptic  spring. 

115.  Materials  for  Springs.  —  The  majority  of  springs  in  com- 
mon use  are  made  from  a  high-grade  steel,  though  frequently 
brass  and  phosphor  bronze  are  found  more  desirable.  In  Chapter 
II  are  given  the  specifications  of  several  grades  of  steel  that  are 
well-adapted  for  the  making  of  springs.  The  permissible  fiber 
stress  varies  with  the  thickness  or  diameter  of  the  material  used 
in  the  construction  of  the  spring,  being  higher  for  the  smaller 
thicknesses  and  diameters  than  for  the  larger.  According  to 
Kimball  and  Barr's  Machine  Design,  the  maximum  allowable 
stress  used  by  an  Eastern  railway  company  in  the  design  of  steel 
leaf  springs  may  be  determined  from  the  following  formula: 


S  =  60,000  +  >  (184) 

in  which  b  represents  the  thickness  of  the  leaves. 


146  MATERIALS  FOR  SPRINGS  [CHAP.  VI 

Quoting  again  from  Kimball  and  Barr,  the  following  formula, 
based  upon  an  experimental  investigation  of  springs  made  in  the 
Sibley  College  Laboratories,  may  be  used  for  arriving  at  the 
probable  working  stress  for  round  stock,  such  as  is  used  in 
the  construction  of  helical  springs: 

Sa  =  40,000  +  ^^  (185) 

in  which  d  represents  the  diameter  of  the  stock. 

The  coefficient  of  elasticity  E  for  all  steels  may  be  assumed  as 
30,000,000,  while  that  for  torsion  or  E8  may  be  taken  at  13,000- 
000. 

The  allowable  working  stresses  and  coefficients  of  elasticity 
for  phosphor  bronze  and  high  brass  spring  stock  are  not  well- 
established,  and  in  the  absence  of  definite  knowledge  relating  to 
the  physical  constants  of  these  materials,  the  following  values 
obtained  from  various  sources  may  be  used: 

For  phosphor  bronze,  S3  varies  from  20,000  to  30,000  pounds 
per  square  inch. 

For  high  brass,  Ss  varies  from  10,000  to  20,000  pounds  per 
square  inch. 

For  high  brass  and  phosphor  bronze  E  =  14,000,000. 

For  high  brass  and  phosphor  bronze  Ea  =    6,000,000. 

In  general,  when  springs  are  subjected  to  vibrations  or  heavy 
shock,  the  stresses  given  above  for  the  various  materials  must  be 
decreased  from  15  to  25  per  cent. 

References 

Elasticitat  und  Festigkeit,  by  C.  BACH. 

Elements  of  Machine  Design,  by  KIMBALL  and  BARR. 

The  Strength  of  Materials,  by  E.  S.  ANDREWS. 

Elements  of  Machine  Design,  by  W.  C.  UNWTN. 

Mechanical  Engineers'  Handbook,  by  L.  S.  MARKS,  Ed.  in  Chief. 

Spring  Engineering,  by  E.  R.  MORRISON. 

Mechanical  Engineers'  Pocket-Book,  by  W.  KENT. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 


CHAPTER  VII 

BELTING  AND  PULLEYS 
BELTING 

The  transmission  of  power  by  means  of  belting  may  be  ac- 
complished satisfactorily  and  efficiently  when  the  distances 
between  the  pulleys  are  not  too  great.  When  the  power  to  be 
transmitted  is  not  large,  round  or  V-shaped  belts  are  used,  the 
latter  form  also  being  used  for  drives  with  short  centers.  The 
materials  used  in  the  construction  of  belting  are  leather,  rubber, 
cotton,  and  steel. 

116.  Leather  Belting. — The  highest  grade  of  leather  belting 
is  obtained  from  the  central  portion  of  the  hide.  This  central 
area  is  cut  into  strips  which  are  cemented,  sewed,  or  riveted  to- 
gether to  form  the  desired  thickness  and  width  of  belt.  The 
thicknesses  vary  from  a  single  hide  thickness  to  that  of  four, 
the  former  being  known  as  a  single  leather  belt  and  the  latter 
as  a  quadruple  belt.  The  terms  double  and  triple  belt  are  used 
when  two  or  three  thicknesses  are  employed  in  the  construction. 
The  hides  from  which  leather  belts  are  made  may  be  tanned  by 
different  processes.  For  ordinary  indoor  installations,  the  regular 
oak-tanned  leather  belting  is  well-adapted.  For  service  in  which 
the  belt  is  exposed  to  steam,  oil  or  water,  a  special  chrome-tanned 
leather  is  recommended.  This  special  tanning  process  is  more 
or  less  secret  and  is  guarded  by  patents.  The  users  of  this 
process  claim  that  a  more  durable  leather  is  produced,  due  to 
the  fact  the  fibrous  structure  of  the  hide  is  preserved  and  not 
weakened  as  may  result  in  the  oak-tanning  process.  Leather 
belting  weighs  on  an  average  about  0.035  pounds  per  cubic 
inch. 

(a)  Commercial  sizes. — Leather  belting  is  made  in  the  follow- 
ing widths: 

From  one-half  to  one  inch,  the  widths  advance  by  J^-inch 
increments. 

147 


148  RUBBER  BELTING  [CHAP.  VII 

From  one  to  four  inches,  the  widths  advance  by  J^-inch 
increments. 

From  four  to  seven  inches,  the  widths  advance  by  J^-inch 
increments. 

From  seven  to  thirty  inches,  the  widths  advance  by  1-inch 
increments. 

From  thirty  to  fifty-six  inches,  the  widths  advance  by  2-inch 
increments. 

From  fifty-six  to  eighty-four  inches,  the  widths  advance  by 
4-inch  increments. 

The  thickness  of  a  single  belt  varies  from  0.16  to  0.25  inch, 
while  that  of  a  double  belt  runs  from  0.3  to  0.4  inch. 

(b)  Strength  of  leather  belting. — The  ultimate  strength  of  oak 
tanned  leather  runs  from  3,000  to  6,000  pounds  per  square  inch, 
the  former  figure  applying  to  the  lower  grades  of  leather  and 
the  latter  to  the  high-grade  product.  According  to  tests  made 
on  chrome-tanned  leather,  the  ultimate  strength  varies  from 
7,500  to  12,000  pounds  per  square  inch.  Table  35  contains  infor- 
mation pertaining  to  the  strength  of  leather  belting,  as  given  by 
Mr.  C.  J.  Morrison,  page  573  of  The  Engineering  Magazine, 
July,  1916. 

117.  Rubber  Belting. — Rubber  belting  is  made  by  fastening 
together  several  layers  of  woven  duck  into  which  is  forced  a 
rubber  composition  which  subsequently  is  vulcanized.  Belting 
of  this  description  is  used  to  some  extent  in  damp  places,  as  for 
example  in  paper  mills  and  saw  mills. 

A  material  resembling  rubber,  known  as  balata,  is  now  used 
extensively  in  the  manufacture  of  an  acid-  and  water-proof  belt. 
Balata  is  made  from  the  sap  of  the  boela  tree  found  in  Venezuela 
and  Guiana.  It  does  not  oxidize  or  deteriorate  as  does  rubber. 
The  body  of  the  belt,  consisting  of  a  heavy  woven  duck,  is  im- 
pregnated and  covered  with  the  balata  gum,  producing  a  belting 
material  which  is  acid-  and  water-proof,  and  according  to  tests 
is  about  twice  as  strong  as  good  leather.  It  is  claimed  that  the 
heating  of  the  belt  due  to  excessive  slippage  softens  the  balata 
and  thereby  increases  its  adhesive  properties.  Due  to  this  fact, 
it  appears  that  balata  belting  is  unsuitable  for  installations  where 
temperatures  of  over  100°F.  prevail. 

The  weight  of  rubber  belting  is  about  0.045  pound  per  cubic 
inch. 


ART.  117] 


LEATHER  BELTING  DATA 


149 


TABLE  35. — RESULTS  OF  TEST  ON  LEATHER  BELTING 


Mfr. 

Sample 

Bel 

Breaking 
strength 

Ultimate 
strength 

Stretch  in  2  inches 

Type 

Size 

Actual 

Per  Cent. 

A 

1 

2 
3 
4 
5 
6 

Double 
Belt 

f 

2X0.406 

2X0.375 
2X0.344 
2X0.3125 

4,000 
3,800 
3,200 
3,430 
3,240 
3,240 

4,930 
4,680 
3,940 
4,575 
4,700 
5,190 

0.25 
0.23 

12.5 
11.5 

0.27 
0.25 
0.22 

13.5 
12.5 
11.0 

7 
8 
9 
10 
11 
12 

Single 
Belt 

2X0.266 

2X0.25 
2X0.219 
2X0.1875 

2,230 
1,880 
2,240 
2,210 
1,840 
2,440 

4,200 
3,540 
4,226 
4,420 
4,200 
6,500 

0.23 
0.21 
0.07 
0.25 
0.23 
Too  small 

11.5 
10.5 
3.5 
12.5 
11.5 
to  measure 

B 

1 
2 
3 
4 

Double 
Belt 

2X0.344  I 
2X0.281  | 

2,280 
2,460 
2,300 
2,310 

3,320 
3,580 
4,100 
4,120 

0.17 
0.27 
0.26 
0.24 

8.5 
13.5 
13.0 
12.0 

5 

6 

7 
8 

Single 
Belt 

2X0.219 
2X0.172 
2X0.1875 
2X0.172 

2,880 
1,700 
1,500 
2,180 

6,550 
4,980 
4,000 
6,380 

Too  small 
0.20 
0.25 
0.18 

to  measure 
10.0 
12.5 
9.0 

C 

1 

Triple 

2X0.50 

4,510 

4,510 

0.45 

22.5 

2 
3 

Double 
Belt 

2X0.4375 
2X0.375 

4,070 
3,010 

4,650 
4,020 

0.30 

15.0 

4 
5 
6 

Single 
Belt 

f 
2X0.250 

2,000 
850 
2,750 

4,000 
1,700 
5,500 

0.25 
0.15 

12.5 
7.5 

D 

1 

2 

Double 
Belt 

2.5X0.344 
2.5X0.3125 

3,920 
3,740 

4,558 
4,800 

0.30 
0.24 

15.0 
12.0 

E 

1 
2 
3 
4 

Double 
Belt 

2X0.344  | 

2X0.50 
2X0.375 

2,730 
2,810 
2,600 
3,240 

3,970 
4,090 
2,600 
4,300 

0.23 
0.20 
0.21 

11.5 
10.0 
10.5 

5 
6 

7 

Single 
Belt 

2X0.188 

2,010 
920 
1,420 

5,360 
2,450 
3,790 

0.20 
0.27 
0.30 

10.0 
13.5 
15.0 

150  TEXTILE  BELTING  [CHAP.  VII 

(a)  Commercial    sizes. — According    to  one   large   rubber-belt 
manufacturer,  the  standard  widths  run  from  1  to  60  inches  as 
follows: 

From  one  inch  to  two  inches,  the  widths  advance  by  J/4-inch 
increments. 

From  two  inches  to  five  inches,  the  widths  advance  by  H-inch 
increments. 

From  five  inches  to  sixteen  inches,  the  widths  advance  by 
1-inch  increments. 

From  sixteen  inches  to  sixty  inches,  the  widths  advance  by 
2-inch  increments. 

The  standard  thicknesses  run  from  two  to  eight  plies. 

(b)  Strength   of  rubber  belting. — Practically  no  experimental 
information  is  available  on  the  strength  of  rubber  belting,  though 
it  is  claimed  by  the  manufacturers  that  a  three-ply  rubber  belt 
is  as  strong  as  a  good  single-thickness  leather  belt.     According 
to  information  obtained  from  the  catalog  of  The  Diamond  Rubber 
Co.,  the  following  values  may  be  used  as  representing  the  net 
driving  tensions  per  inch  of  width  for  a  rubber  belt  having  an  arc 
of  contact  of  180  degrees. 

For  a  three-ply  belt  use  40  pounds  per  inch  of  width. 

For  a  four-  and  five-ply  belt  use  50  pounds  per  inch  of  width. 

For  a  six-ply  belt  use  60  pounds  per  inch  of  width. 

For  a  seven-ply  belt  use  70  pounds  per  inch  of  width. 

For  an  eight-ply  belt  use  80  pounds  per  inch  of  width. 

For  a  ten-ply  belt  use  120  pounds  per  inch  of  width. 

118.  Textile  Belting. — Textile  belts  are  made  by  weaving  them 
in  a  loom  or  building  them  up  of  layers  of  canvas  stitched 
together.  The  woven  body  or  strips  of  canvas  are  treated  with 
a  filling  to  make  them  water-proof,  and  in  some  cases  oil-proof. 
Generally,  belts  treated  with  a  cheap  filling  are  very  stiff  and 
hence  do  not  conform  to  the  pulley,  making  it  more  difficult 
to  transmit  the  desired  power.  Textile  belts  are  used  more  for 
conveyor  service  than  for  the  transmission  of  power. 

(a)  Commercial  sizes. — The  sizes  of  oiled  and  stitched  duck 
belting  are  as  follows: 

Four-ply  is  made  in  widths  from    1  inch     to  48  inches. 

Five-  and  six-ply  are  made  in  widths  from    2  inches  to  48  inches. 

Eight-ply  is  made  in  widths  from    4  inches  to  48  inches. 

Ten-ply  is  made  in  widths  from  12  inches  to  48  inches. 

From  one  to  five  inches,  the  widths  vary  by  %-mch  incre- 


ART.  119]  STEEL  BELTING  151 

ments;  from  five  to  sixteen,  by  1-inch  increments;  and  from  six- 
teen to  forty-eight,  by  2-inch  increments. 

White  cotton  belting  is  made  in  the  following  sizes: 

Three-ply  having  a  width  from  1^  inches  to  24  inches. 

Four-ply  having  a  width  from  2      inches  to  30  inches. 

Five-ply  having  a  width  from  4      inches  to  30  inches. 

Six-ply  having  a  width  from  6      inches  to  30  inches. 

Eight-ply  having  a  width  from  6      inches  to  30  inches. 

The  widths  of  the  cotton  belting  vary  as  follows :  from  one  and 
one-half  to  six  inches,  by  J-^-inch  increments ;  from  six  to  twelve, 
by  1-inch  increments;  and  from  twelve  to  thirty,  by  2-inch 
increments. 

119.  Steel  Belting. — The  transmission  of  power  by  means  of 
steel  belts  was  first  introduced  in  1906  by  the  Eloesser  Steel  Belt 
Co.  of  Berlin,  Germany,  and  at  the  present  time  this  method  of 
transmitting  power  is  recognized  by  many  German  engineers  as 
being  superior  to  that  in  which  leather  belting  or  ropes  are  used. 

The  steel  belt  is  used  in  the  same  manner  as  the  leather  belt, 
except  that  it  is  narrow,  thin  and  of  very  light  weight.  It  is 
put  on  the  pulley  with  a  fairly  high  initial  tension  and  hence  runs 
without  sag.  The  material  used  in  making  steel  belts  is  a  char- 
coal steel,  prepared  and  hardened  by  a  secret  process.  After 
rough  rolling  at  a  red  heat,  the  metal  band  is  allowed  to  cool  and 
later  is  finished  to  exact  size.  The  thicknesses  vary  from  0.2 
to  1  millimeter  (0.0079  to  0.039  inch),  and  the  widths  range  from 
30  to  200  millimeters  (1.18  to  7.87  inches).  The  ultimate  tensile 
strength  of  the  finished  material  is  approximately  190,000  pounds 
per  square  inch. 

The  pulleys  upon  which  these  belts  run  are  preferably  flat,  and 
are  covered  with  layers  of  canvas  and  cork  so  as  to  increase  the 
coefficient  of  friction.  A  crowned  pulley  may  be  used,  provided 
the  crown  does  not  exceed  approximately  33  ten-thousandths  of 
the  width  of  the  belt.  Steel  belts  are  not  adapted  to  tight  and 
loose  pulleys,  but  crossed  belts  will  work  satisfactorily,  provided 
the  distance  between  the  shafts  is  about  seventy  times  the  width 
of  the  belt. 

In  case  the  power  transmitted  is  large,  so  that  a  single  belt  of 
sufficient  width  to  give  the  required  cross-sectional  area  cannot 
be  obtained,  two  or  more  belts  are  run  side  by  side.  In  putting 
steel  belts  on  pulleys,  a  special  clamp  is  used  in  order  to  measure 


152  STEEL  BELTING  [CHAP.  VII 

correctly  the  initial  tension  and  at  the  same  time  to  facilitate 
fitting  the  special  plates  necessary  to  make  the  joint.  The  de- 
sign of  a  proper  fastening  for  steel  belts  presented  a  difficult 
problem,  but  after  considerable  experimental  work  D.  Eloesser, 
now  head  of  the  firm  that  bears  his  name,  perfected  a  joint  that 
has  proven  very  satisfactory.  His  first  design  was  made  of  one 
piece  and  the  ends  of  the  belt  had  to  be  soldered  in  place  at  the 
installation.  The  latest  design,  shown  in  Fig.  57,  consists  of 
several  parts  fastened  together  by  screws  e  that  are  removable. 
The  ends  of  the  steel  band  are  soldered  to  the  main  parts  of  the 
joint  and  the  small  screws/ and  g  passing  through  the  triangular- 
shaped  steel  pieces  c  and  d  give  added  strength  to  the  fastening. 
The  plates  a  and  b  that  form  the  main  parts  of  the  joint  are 
curved,  the  curvature  depending  upon  the  size  of  the  pulley 
upon  which  the  belt  is  to  run. 

(b)  Experimental  conclusions^ — The  following  conclusions  were 
derived  from  a  study  of  a  large  number  of  tests  on  steel  belts 
made  in  actual  service. 


FIG.  57. 

1.  Steel  belts  do  not  stretch  after  being  placed  on  the  pulleys, 
hence  there  is  no  necessity  for  taking  up  slack. 

2.  Steel  belts  are  not  affected  by  variations  in  temperature  and 
may  be  used  satisfactorily  in  damp  places. 

3.  Steel  belts  will  transmit  the  same  horse  power  as  leather 
belts  having  a  width  two  to  four  times  as  great. 

4.  Due  to  the  decrease  in  width  over  leather  belts  transmitting 
the  same  power,  narrower-face  pulleys  may  be  used,  thus  effect- 
ing a  considerable  saving  in  the  cost  of  the  pulley  and  in  space 
due  to  a  reduction  in  the  general  dimensions  of  machinery. 

5.  It  is  claimed  that  the  first  cost  of  steel  belting  is  less  than 
that  of  leather  or  rubber  belting. 

6.  Steel  belts  are  more  sensitive  and  hence  the  pulleys,  as  well 
as  the  shafting,  require  more  accurate  alignment. 

7.  Speeds  as  high  as  19,500  feet  per  minute  have  been  attained, 
and  the  slip  at  this  speed  was  only  0.15  of  1  per  cent. 


ART.  119]  STEEL  BELTING  153 

8.  Due  to  the  small  slip,  steel  belts  transmit  power  virtually 
without  loss. 

9.  Steel  belts  do  not  wear,  and,  if  properly  installed,  are  said 
to  have  a  useful  life  exceeding  five  years. 

10.  As  the  tension  in  steel  belts  is  only  a  fraction,  about 
one-tenth,  of  that  used  in  a  leather  belt  of  the  same  capacity,  the 
pressures  on  the  bearings  are  less,  thus  reducing  the  frictional 
losses. 

11.  Steel  belts  weigh  much  less  than  leather  belts  of  equal 
capacity,  and  hence  reduce  the  frictional  losses  still  more. 

12.  Due  to  the  extreme  thinness  of  steel  belts  and  the  high 
speeds  used,  they  might  prove  dangerous  if  the  drive  is  not  en- 
closed by  proper  guards. 

(c)  Results  of  tests. — The  following  results,  collected  from  the 
various  reports  recorded  in  several  German  technical  journals, 
are  given  to  show  what  actually  has  been  accomplished  in  the 
transmission  of  power  by  means  of  steel  belting. 

1.  Under  ordinary  running  conditions,  a  4-inch  steel  belt  is 
equivalent  to  an  18-inch  leather  belt  or  six  manila  ropes  1%  inches 
in  diameter. 

2.  In  a  particular  installation,  a  4-inch  steel  belt  transmitted 
250  horse  power,  having  replaced  a  24-inch  leather  belt. 

3.  Two  steel  belts  each  5.9  inches  wide  were  used  to  transmit 
450  horse  power,  which  formerly  required  12  cables. 

4.  A  6-inch  steel  belt  0.024  inch  thick  is  capable  of  transmitting 
200  horse  power,  and  with  two  such  belts  placed  side  by  side  on 
the  same  pulley,  440  horse  power  has  been  transmitted. 

5.  Three  4%-inch  steel  belts  were  used  to  transmit  1300  horse 
power  at  500  revolutions  per  minute  of  the  driven  pulley.     The 
distance  between  the  122-inch  driving  and  63-inch  driven  pulleys 
was  46  feet. 

6.  In  another  installation,  75  horse  power  was  transmitted  by 
a  6-inch  steel  belt  running  over  pulleys  108  and  51  inches  in 
diameter,  located  on  76-inch  centers. 

(d)  American  experiments  on  steel  belting. — In  1911  or  1912, 
the  General  Electric  Co.  made  a  series  of  experiments  with  steel 
belts,  and  came  to  the  conclusion  that  they  were  not  entirely 
satisfactory.     The  thicknesses  of  the  belts  used  in  these  experi- 
ments varied  from  0.007  to  0.018  inch.     A  %-inch  belt  0.01 
inch  thick  was  capable  of  transmitting  150  horse  power  con- 
tinuously for  17  hours  at  a  speed  of  20,000  feet  per  minute.     This 


154  BELT  FASTENERS  [CHAP.  VII 

belt  was  made  of  cold-rolled  steel  and  the  initial  tension  put 
on  the  belt  in  order  to  give  the  above  results  was  90,000  pounds 
per  square  inch.  The  General  Electric  Co.  found  that  steel  belts 
will  not  run  satisfactorily  on  the  ordinary  steel  pulleys,  and  the 
best  results  were  obtained  with  a  leather-faced  pulley.  No  doubt 
the  following  are  some  of  the  reasons  why  the  results  obtained 
by  the  General  Electric  Co.  from  their  investigation  on  steel 
belting  were  not  as  promising  as  those  found  by  the  German 
engineers: 

1.  Not  as  good  a  grade  of  steel  available  for  making  the  band. 

2.  Probably  during  the  early  stages  of  preparing  the  band,  im- 
proper treatment  gave  rise  to  scale  troubles. 

3.  Difficulty  in  the  process  of  annealing. 

4.  Lack  of  time  for  further  research  work. 

120.  Belt  Fastenings. — Fastenings  of  various  forms  are  used 
for  joining  the  ends  of  a  belt,  but  none  of  them  is  as  strong  and 
durable  as  the  scarfed  and  glued  splice,  which  when  made  care- 
fully is  but  little  weaker  than  the  belt  proper.  Of  necessity,  the 
scarfed  and  glued  joint  or  cemented  splice  is  adapted  to  installa- 
tions in  which  the  slack  of  the  belt  is  taken  up  by  mechanical 
means,  and  where  careful  attention  is  given  to  belting  by  com- 
petent workmen.  Probably  the  oldest  form  of  fastening,  as  well 
as  that  used  most  commonly,  is  to  join  the  ends  of  a  belt  by 
means  of  rawhide  lacing.  Not  infrequently  belts  are  laced  to- 
gether with  wire,  and  such  joints  run  very  smoothly,  especially 
if  made  with  a  machine,  and  are  considerably  stronger  than  the 
rawhide  laced  joint,  as  is  indicated  in  Table  36.  Patented 
metal  fasteners  in  the  form  of  hooks,  studs,  and  plates  are  also 
in  use  and  have  the  advantage  that  they  are  cheap  and  applied 
very  easily  and  quickly.  Some  of  the  metal  fasteners  are  too 
dangerous  to  be  used  on  belts  that  must  be  touched  by  hand,  and 
for  that  reason  some  states  have  legislated  against  their  use. 

Tests  of  belt  joints. — Tests  of  various  types  of  belt  joints 
were  made  at  the  University  of  Wisconsin,  also  at  the  University 
of  Illinois.  In  The  Engineering  Magazine  of  July,  1916,  Mr.  C. 
J.  Morrison  presented  a  valuable  article  entitled  "Belts — Their 
Selection  and  Care, "  in  which  he  gives  considerable  information 
pertaining  to  the  strength  of  leather  belts  and  the  joints  used 
with  such  belting.  In  Table  36  is  given  information  pertaining 
to  the  strengths  and  efficiencies  of  the  various  types  of  leather 
belt  joints  tested  by  Mr.  Morrison.  It  should  be  understood 


ART.  121] 


BELT  TENSIONS 


155 


that  the  term  " efficiency"  in  this  case  is  used  in  the  sense  as 
when  applied  to  riveted  joints. 

TABLE  36. — STRENGTH  OF  LEATHER  BELT  JOINTS 


Type  of  joint 

Breaking 
load, 
pounds 

Efficiency, 
per  cent. 

Cemented 
splice 

Wire,  machin 
Wire,  hand-k 
Rawhide  wit' 
Rawhide  wit 
Metal  hooks. 

Cement  only  

2,440 
2,430 
2,170 
2,060 
2,040 
5,850 
5,330 
4,100 
3,200 
2,270 
1,950 

100.0 
99.6 
88.9 
84.4 
83.6 
90.0 
82.0 
63.0 
49.0 
35.0 
30.0 

Cement  and  shoe  pegs. 

Cement  and  small  copper  rivets  .  .  . 
Cement  and  small  copper  rivets  .  .  . 
Cement  and  large  copper  rivets  
e-laced                                        .    .  . 

iced  

i  small  holes                             

i  large  holes 

Metal  studs 

STRESSES  IN  BELTING 

121.  Tensions  in  Belts. — A  belt  transmits  power  due  to  its 
friction  upon  the  face  of  the  pulley.  This  transmitting  capacity 
depends  upon  the  following  important  factors: 

(a)  The  allowable  net  tension  in  the  belt. 

(6)  The  coefficient  of  friction  existing  between  the  belt  and 
pulley. 

(c)  The  speed  at  which  the  belt  is  running. 

Net  tensions. — The  net  tension  represents  the  capacity  of 
the  belt  and  depends  upon  the  maximum  allowable  tension, 
the  coefficient  of  friction,  the  angle  of  contact  that  the  belt 
makes  with  the  pulley,  the  material  of  both  the  belting  and  the 
pulley,  the  diameter  of  the  pulley,  and  the  velocity  of  the  belt. 
The  net  tension  is  not  a  constant  as  is  frequently  assumed,  but 
it  varies  with  the  speed.  Let  two  pulleys  be  connected  by  a 
belt  as  shown  in  Fig.  58,  and  assume  that  no  power  is  being  trans- 
mitted, except  that  required  to  overcome  the  frictional  resistance 
on  the  bearings  due  to  the  initial  tension  with  which  the  belt  was 
placed  on  the  pulleys.  Due  to  this  initial  tension,  which  is  the 
same  on  both  the  running  on  and  off  sides  of  the  pulleys,  the  belt 
exerts  a  pressure  upon  the  face  of  the  pulleys.  This  pressure  in 
turn  induces  a  frictional  force  on  the  rim  capable  of  overcoming 


156 


RATIO  OF  BELT  TENSIONS 


[CHAP.  VII 


an  equivalent  resistance,  tending  to  produce  relative  motion 
between  the  belt  and  pulley.  The  tensions  in  the  two  parts 
of  the  belt  will  change  as  soon  as  power  is  transmitted,  say  from 
a  to  b,  causing  that  in  the  pulling  side  to  increase  and  that  in  the 
running  off  side  to  decrease.  Representing  these  tensions  by  the 
symbols  T\  and  T2,  we  see  that  the  force  causing  the  driven  pulley 
b  to  rotate  is  the  difference  of  these  tensions,  or  T\  —  TV  This 
difference  is  known  as  the  net  tension. 

It  is  evident  that  due  to  this  difference  in  tension  in  the  various 
sections  of  the  belt,  a  unit  length  of  the  belt  in  running  from  the 
point  A  to  B,  decreases  in  length  due  to  its  elasticity.  From 
this  it  follows  that  the  driver  a  delivers  a  shorter  length  of  belt  at 


FIG.  58. 

B  than  it  receives  at  A  and  furthermore,  that  the  velocity  of  the 
pulley  face  and  that  of  the  belt  are  not  equal.  A  similar  action 
occurs  on  the  pulley  b.  This  action  is  known  as  belt  creep  and 
results  in  some  loss  of  power. 

122.  Relation  between  Tight  and  Loose  Tensions. — The 
horse  power  delivered  by  a  belt  may  be  determined  as  soon  as 
the  net  tension  and  the  speed  are  established;  hence  it  is  im- 
portant to  derive  the  relation  existing  between  the  tight  and 
loose  tensions. 

Let  A  =  cross-sectional  area  of  the  belt  in  square  inches. 
C  =  centrifugal  force  of  an  elementary  length  of  belt. 

5  =  allowable  working  stress  of  the  belt. 
b  =  width  of  belt.  » 

t  =  thickness  of  belt. 
v  =  velocity  of  belt,  in  feet  per  second. 
w  —  weight  of  belt,  pounds  per  cubic  inch, 
ju  =  coefficient  of  friction. 

6  =  total  angle  of  contact,  expressed  in  radians. 


ART.  122] 


RATIO  OF  BELT  TENSIONS 


157 


In  Fig.  59  a  short  portion  of  the  belt  has  an  arc  of  contact  sub- 
tending the  angle  A0  at  the  center  of  the  pulley.  Let  the  tension 
at  one  end  be  T  and  at  the  other  (T  +  AT1);  evidently  each  of 

these  tensions  makes  an  angle  I  ~ o|  w^n  ^ne  vertical  center 

line.  The  pressure  between  the  portion  of  the  belt  and  the  pulley 
rim  is  designated  by  the  symbol  N,  and  the  force  of  friction 
between  them  is  /JV.  In  addition  to  these  forces,  we  have  the 
centrifugal  force  C  acting  radially  as  shown  in  the  figure.  The 
magnitude  of  the  centrifugal  force  is  given  by  the  following 
expression : 

(186) 


T+AT 


FIG.  59. 


The  piece  of  belt  referred  to  above  is  held  in  equilibrium  by 
the  five  forces  T,  (T  +  AT7),  N,  /JV,andC.  The  summation  of 
the  horizontal  and  vertical  components,  respectively,  gives  the 
following  equations : 


-  AT7  cos  ~  + 


=  0 


(187) 


(2  T  +  AT7)  sin      -  -  N  -  C  =  0 


(188) 


Eliminating  N 


M  (2  T  +  AT)  sin  A?  -  AT  cos  -^  - 

4  & 


158  RATIO  OF  BELT  TENSIONS  CHAP.  VII] 

Dividing  through  by  -^-,  and  passing  to  the  limit,  we  get 

A0 

ClTl     __ 

r       /nm    i    Aimr  2          24  »Awv*        or        A^  r  A<? 

/u  lim  (2  T  +  AT7)  lim  —  —  ---  —     -  =  2  hm  —  -  lim  cos  — 

At7  (J  iA(7  J 

"T 

whence 

^  =  M  (r  -  *)  (189) 

where 


12Awv* 

k  =  - 


Separating  the  variables 


Integrating,  we  find  that  the  relation  between  the  tight  and  loose 
tensions  is  as  follows: 


From  (190),  we  find  that  the  net  tension  is 

r,  -  r,  -  (T,  -  k)  (i9i) 


Substituting  in  (191)  the  value  of  7\  in  terms  of  b,  2  and  S,  we 
have 

T        T  12 

Tl~      ~- 


Denoting  the  terms     S  --      and     -  —  ^  —     by  the   sym- 
bols m  and  n,  respectively,  we  get  finally 

Tl  -  T2  =  mnU  (192) 

Having  determined  the  magnitude  of  the  net  tension  from 
(192)  and  knowing  the  speed  v,  the  horse  power  delivered  may  be 
calculated  from  the  relation 

H  =  5^6  (Tl  ~  Tz)  (193) 

123.  Coefficient  of  Friction.  —  There  is  much  diversity  of 
opinion  regarding  the  working  coefficient  of  friction,  but  in  general 
it  depends  upon  the  material  of  the  belt  and  the  condition  of  the 


ART.  124] 


COEFFICIENT  OF  FRICTION 


159 


belt,  the  permanent  slip,  whether  the  load  is  steady  or  fluctuat- 
ing, the  diameter  of  the  pulley  and  the  material  of  which  it  is 
made,  and  the  speed  of  the  belt.  In  view  of  the  foregoing,  the 
coefficient  of  friction  cannot  be  assumed  as  an  average  for  all 
speeds,  as  is  so  frequently  done  in  belting  calculations.  It  is 
practically  impossible  to  derive  an  expression  for  ju  in  terms  of 
all  of  the  factors  mentioned  above,  but  the  following  formula 
proposed  by  Mr.  C.  G.  Barth  has  been  found  to  give  fairly  satis- 


—0.55 


5000 


Speed    of    Belt -ft  per  min. 
4000  3000  2000 


0-45 


0.40 


030 


—0.25- 


0          100'        200        500        400        500        600        700        800        900        1000 
Speed   of  5elt  -ft  per  min. 
FIG.  60. 

factory  results  in  practice  for  leather  belting  on  cast-iron  or 
steel-rim  pulleys. 

=  0.54  -  _14?  _-,  (194) 


in  which  V  represents  the  velocity  of  the  belt  in  feet  per  minute 
The  Barth  formula  for  ju,  as  given  by  (194),  has  been  evaluated 

for  various  values  of  7,  and  the  results  obtained  are  shown  in 

graphic  form  in  Fig.  60. 

124.  Maximum  Allowable  Tension. — The  maximum  allowable 

tension  that  may  be  put  upon  a  belt  depends  upon  the  quality  of 

the  material,  the  permanent  stretch  of  the  belt,  the  imperfect 


160 


SELECTION  OF  BELT  SIZE 


[CHAP.  VII 


elasticity  of  the  belting  material,  and  the  strength  of  the  joints 
in  the  belts.  In  Table  37  are  given  the  average  values  for 
the  ultimate  strengths  of  leather  belting,  as  given  by  Morrison 
in  his  article  referred  to  previously.  To  arrive  at  the  magnitude 
of  the  allowable  working  stress  S  for  leather,  multiply  the  ulti- 
mate strength  by  the  so-called  efficiency  of  the  joint  and  divide 
the  product  thus  obtained  by  the  assumed  factor  of  safety.  As 
an  aid  in  the  solution  of  belt  problems,  the  several  factors  just 
mentioned,  as  well  as  the  allowable  working  stresses  for  the  im- 
portant joints  used  in  connection  with  leather  belting,  are  given 
in  Table  38. 

TABLE  37. — AVERAGE    ULTIMATE    STRENGTH    OF   LEATHER    BELTING 


Mfr. 

No.  of 
samples 

Best 

Poorest 

Average 

Remarks 

A 

12 

6,500 

3,549 

4,611 

B 

8 

6,550 

3,303 

4,614 

( 

6 

5,500 

1,700 

4,062 

Poorest  broke  in  the 

C 

splice. 

I 

5 

5,500 

4,013 

4,532 

Omitting  1,700   sam- 

ple. 

D 

2 

4,800 

4,558 

4,679 

E 

7 

5,360 

2,453 

3,800 

TABLE  38. — WORKING  STRESSES  FOR  LEATHER  BELTING 


Type  of  joint 

Ultimate 
strength 

Efficiency 
of  joint 

Factor  of 
safety 

Working 
stress  S 

Cemented  

0.98 

420 

f  Machine-laced.  . 
Wire  <  „      ,  . 
{  Hand-laced  

4,300 

0.88 
0.80 

10 

380 
340 

Rawhide  laced  

0.60 

260 

125.  Selection  of  Belt  Size. — Having  arrived  at  the  allowable 
working  stress  in  a  belt,  and  knowing  the  magnitude  of  the  net 
driving  tension  P  as  well  as  the  angle  of  contact  B  and  the  coef- 
ficient of  friction  /*,  the  area  of  the  belt  may  be  calculated  by 
means  of  (192).  From  the  conditions  of  the  problem,  either 
the  width  of  the  belt  or  its  thickness  may  be  established;  hence 
the  remaining  dimension  may  be  determined.  Now  the  selec- 
tion of  the  proper  belt  thickness  is,  in  general,  determined  by  the 
diameter  of  the  smallest  pulley  used  in  the  transmission,  If  the 


ART.  126]  TAYLOR'S  EXPERIMENTS  161 

belt  is  thick  relative  to  the  diameter  of  the  smallest  pulley,  the 
result  will  be  an  unsatisfactory  drive,  due  to  the  excessive  slip- 
page and  belt  wear,  as  well  as  the  excessive  loss  of  power.  In 
addition  to  the  points  just  mentioned,  the  result  of  running  a 
thick  belt  over  a  small  pulley  will  be  a  considerable  decrease  in 
the  life  of  the  belt. 

Satisfactory  belt  service,  as  well  as  long  life,  is  secured  if  the 
diameter  of  the  smallest  pulley  in  the  transmission  is  made  not  less 
than  12  inches  if  a  double  belt  of  medium  or  heavy  weight  is 
used ;  for  a  triple  belt,  the  minimum  diameter  of  pulley  should  be 
20  inches,  and  for  a  quadruple  belt,  30  inches.  The  selection  of  a 
belt  thickness  may  also  be  influenced  to  a  certain  degree  by  the 
fact  that  good  reliable  single  belts  are  hard  to  obtain  in  widths 
exceeding  12  to  15  inches.  A  rule  occasionally  used  for  the  limit- 
ing size  of  a  single  belt  is  as  follows:  "  A  single  belt  should  never 
be  used  where  the  width  is  more  than  four-thirds  the  diameter  of 
the  smallest  pulley." 

126.  Taylor's  Experiments  on  Belting. — In  volume  XV  of  the 
Transactions  of  the  American  Society  of  Mechanical  Engineers, 
Mr.  F.  W.  Taylor  reports  "A  Nine  Years'  Experiment  on  Belt- 
ing" carried  on  at  the  Midvale  Steel  Co.  This  paper  gives  some 
valuable  data  on  the  actual  performance  of  belts,  and  a  satis- 
factory abstract  of  it  is  impossible  in  this  chapter.  The  conclu- 
sions, thirty-six  in  number,  given  in  the  paper  are  based  upon  the 
cost  of  maintaining  the  belts  in  good  condition,  including  time 
lost  in  making  repairs,  as  well  as  other  considerations.  The 
following  are  some  of  the  conclusions: 

(a)  Thick  narrow  belts  are  more  economical  than  thin  wide 
ones. 

(b)  The  net  driving  tension  of  a  double  belt  should  not  exceed 
35  pounds  per  inch  of  width,  but  the  initial  tension  may  be  double 
that  value. 

(c)  The  most  economical  belt  speed  ranges  from  4,000  to 
4,500  feet  per  minute. 

(d)  For  pulleys  12  inches  in  diameter  or  larger  double  belts 
are  recommended. 

For  pulleys  20  inches  in  diameter  or  larger  triple  belts  are 
recommended. 

For  pulleys  30  inches  in  diameter  or  larger  quadruple  belts 
are  recommended. 


162  TANDEM-BELT  TRANSMISSION  [CHAP.  VII 

(e)  The  joints  should  be  spliced  and  cemented  rather  than 
laced  with  rawhide  or  wire,  or  joined  by  studs  or  hooks. 

(/)  Belts  should  be  cleaned  and  greased  every  five  or  six 
months. 

(g)  The  best  distance  between  centers  of  shafts  is  from  twenty 
to  twenty-five  feet. 

(ti)  The  face  of  a  pulley  should  be  25  per  cent,  wider  than  the 
belt. 

127.  Tandem-belt  Transmission. — Not  infrequently  two  belts, 
one  placed  on  top  of  the  other,  are  used  to  transmit  power  from 
one  pulley  to  two  separate  pulleys.     This  arrangement  is  known 
as  a  tandem-belt  drive.     The  outside  belt  travels  at  a  somewhat 
higher  speed  than  the  inner,  and  this  fact  must  not  be  lost  sight 
of  when  a  tandem-belt  transmission  is  being  designed  in  which 
the  speeds  of  the  two  driven  pulleys  must  be  the  same.     Experi- 
ence with  tandem-belt  drives  has  shown  that  the  best  results 
are  obtained  when  both  belts  are  of  the  same  thickness,  prefer- 
ably of  double  thickness,  and  are  placed  upon  the  pulleys  with 
the  same  initial  tension.     Due  to  the  higher  coefficient  of  fric- 
tion between  leather  and  leather,  practically  all  the  slip  will 
occur  between  the  pulley  and  the  inner  belt.     To  arrive  at  the 
proper  size  of  a  belt  required  for  a  tandem  drive,  proportion  each 
belt  according  to  the  power  it  must  transmit. 

128.  Tension  Pulleys. — Whenever  possible,  it  is  well  to  provide 
means  of  releasing  the  initial  tension  in  belts  during  extended 
periods  of  idleness.     In  some  cases,  as  in  electrical  machinery, 
this  is  accomplished  by  mounting  the  machines  on  rails,  thus 
providing  means  for  changing  the  distance  between  the  centers 
of  the  pulleys.     To  a  certain  extent,  the  practice  of  making  the 
loose  pulley  on  machine  drives  smaller  in  diameter,  will  relieve 
the  belt  tensions.     There  are,  however,  many  belting  installations 
where  neither  of  these  methods  could  be  used,  and  in  many  of 
these  cases  tension  pulleys  designed  and  installed  properly  will 
improve  the  transmission. 

Lenix  system. — In  the  Lenix  system,  the  tension  pulley  is 
placed  on  the  slack  side  of  the  belt  as  near  to  the  smaller  pulley 
in  the  transmission  as  is  practicable.  The  general  features  of 
this  system  are  shown  in  the  two  radically  different  installations 
represented  in  Figs.  61  and  62.  The  tension  pulley  is  carried  on 
an  arm  pivoted  on  the  axis  of  the  small  driving  pulley,  and  by 


ART.  128] 


TENSION  PULLEYS 


163 


means  of  a  weight  the  required  tension  may  be  put  on  the  slack 
belt.  In  the  installation  shown  in  Fig.  61,  the  tension  on  the 
belt  is  changed  by  increasing  or  decreasing  the  leverage  of  the 


FIG.  61. 


tension  weight.  It  is  evident  from  an  inspection  of  Figs.  61  and 
62,  that  a  large  arc  of  contact  is  obtained  by  means  of  this  system 
and  for  that  reason  the  tension  in  the  belt  may  be  reduced. 


W///////M 


FIG.  62. 


The  diameter  of  the  tension  pulley  should  never  be  made  less 
than  that  of  the  smallest  pulley  in  the  drive.     The  only  losses 


164 


PULLEYS 


[CHAP.  VII 


chargeable  to  the  tension  pulley  are  those  due  to  journal  friction, 
which,  if  the  apparatus  is  properly  designed  and  erected,  are 
small  and  have  practically  no  effect  on  the  efficiency  of  the  trans- 
mission. Some  additional  advantages  of  tension  pulleys  are  as 
follows:  (1)  the  initial  tension  of  the  belt  may  be  regulated  very 
accurately  and  may  be  maintained  at  the  proper  magnitude;  (2) 
during  periods  when  the  drive  is  not  in  use  the  belt  may  be  re- 
lieved of  the  initial  tensions. 


PULLEYS 

129.  Types  of  Pulleys. — (a)  Cast-iron  pulleys. — Pulleys  are 
made  from  various  kinds  of  materials,  cast  iron,  however,  being 
the  most  common.  As  far  as  the  cost  of  manufacture  is  con- 
cerned, cast  iron  is  ideal  since  it  can  be  cast  in  any  desired  shape, 
though  precautions  must  be  taken  in  the  foundry  when  light- 


FIG.  63. 

weight  pulleys  are  cast.  If  the  metal  in  the  various  parts  of  the 
pulley  is  not  distributed  correctly,  shrinkage  stresses  due  to 
irregular  cooling  are  likely  to  reduce  the  useful  strength  of  the 
material.  To  partly  overcome  this  trouble,  pulleys  are  split  in 
halves.  Careless  moulding  in  the  foundry  generally  produces 
pulleys  having  rims  that  are  not  uniform  in  thickness,  thus  caus- 
ing them  to  run  out  of  balance.  This  defect  is  rather  serious  in 
a  high-speed  transmission,  though  the  pulley  can  be  balanced  by 
attaching  weights  at  the  lightest  points.  The  centrifugal  force 


ART.  129] 


PULLEYS 


165 


due  to  these  weights  will  set  up  severe  stresses  in  the  weak  rim 
and  may  cause  it  to  burst. 

(b)  Steel  pulleys. — A  type  of  pulley  introduced  to  overcome 
some  of  the  defects  of  cast-iron  pulleys  consists  of  a  cast-iron 
hub  and  arms  to  which  is  riveted  a  steel  rim.  Pulleys  built  in 
this  way  are  lighter  than  cast-iron  ones  for  the  same  duty,  but 
trouble  may  result  with  the  fastenings  as  they  may  work  loose 
due  to  the  heavy  loads  transmitted.  Pulleys  built  entirely  of 
steel  are  also  used,  and  are  looked  upon  with  favor  by  many 
engineers.  In  Figs.  63  and  64  are  shown  the  designs  of  a  small 
and  large  pulley  as  manufactured  by  The  American  Pulley  Co. 
of  Philadelphia.  An  inspection  of  Figs.  63  and  64  shows  that 
the  construction  adopted  for  these  pulleys  gives  a  maximum 
strength  for  a  minimum  weight,  and  furthermore,  the  windage 
effect  at  high  speeds  is  small. 


FIG.  64. 

(c)  Wood  pulleys. — Wood  pulleys  in  the  smaller  sizes  generally 
consist  of  a  cast-iron  hub  upon  which  is  fastened  a  wood  rim  built 
up  of  segments  of  well-seasoned  maple.     In  the  larger  sizes,  they 
are  always  made  in  the  split  form  and  are  built  entirely  of  wood. 
Due  to  atmospheric  conditions,  wood  pulleys  are  very  likely  to 
warp  or  distort,  which  may  cause  trouble  at  high  speeds. 

(d)  Paper  pulleys. — Pulleys  made  of  paper  are  also  in  com- 
mon use.     As  shown  in  Fig.  65,  such  a  pulley  consists  of  a  web 
and  rim  built  up  of  thin  sheets  of  straw  fiber  cemented  together 
and  compressed  under  hydraulic  pressure.     To  secure  additional 
strength  in  the  rims,  wooden  dowel  pins  extend  through  the  rim 
and  web  as  shown  in  the  figure.     The  webs  are  clamped  securely 
between  the  flanges  of  the  cast-iron  hub  as  shown. 

(e)  Cork  insert  pulleys. — Frequently  pulleys  are  lagged  with 


166 


TRANSMITTING  CAPACITY  OF  PULLEYS    [CHAP.  VII 


leather  or  cotton  belting  in  order  to  increase  the  coefficient  of 
friction  between  the  belt  and  pulley.  However,  such  lagging 
wears  out  quickly  and  must  be  renewed,  thus  increasing  materi- 
ally the  cost  of  upkeep  of  the  transmission.  It  has  been  found 
by  an  extended  series  of  experiments,  conducted  by  Prof.  W.  M. 
Sawdon  of  Cornell  University,  that  the  transmitting  capacity  of 
practically  any  type  of  pulley  can  be  increased  by  fitting  cork 
inserts  into  the  face.  The  corks  are  pressed  into  the  face  and 
allowed  to  protrude  above  the  surface  of  the  material  of  the  face 
not  to  exceed  J^2  inch.  These  cork  inserts  dc  not  wear  down 
nearly  as  rapidly  as  the  lagging;  however,  the  first  cost  is  con- 
siderably more. 


FIG.  65. 

130.  Transmitting  Capacity  of  Pulleys. — In  September,  1911, 
before  the  National  Association  of  Cotton  Manufacturers,  Prof. 
W.  M.  Sawdon  read  a  paper  entitled  "  Tests  of  the  Transmitting 
Capacities  of  Different  Pulleys  in  Leather  Belt  Drives,"  in  which 
he  presented  the  results  of  an  extended  investigation  on  the  trans- 

TABLE  39. — COMPARATIVE  TRANSMITTING  CAPACITIES  OF  PULLEYS 


Type  of  pulley 

Relative  capacities  at  various 
slips 

1 
per  cent. 

1H 

per  cent. 

2 
per   cent. 

1   Cast  iron. 

100.0 
133.5 

139.3 
136.3 
130.7 
130.7 
160.7 
149.0 
150.2 

100.0 

119.7 
124.1 
118.8 
116.8 
118.2 
151.7 
135.5 
145.3 

100.0 
107.0 
112.0 
105.6 
104.8 
104.8 
137.3 
122.0 
133.0 

2  Cast  iron  with  corks  projecting  0.04  inch. 
3   Cast  iron  with  corks  projecting  0.015  inch.  . 
4  Wood 

5  Wood  with  corks  projecting  0.075  inch  
6  Wood  with  corks  projecting  0  03  inch  .. 

7  Paper  

8  Paper  with  corks  projecting  0.087  inch  
9  Paper  with  corks  projecting  0.015  inch  

ART.  131] 


PROPORTIONS  OF  PULLEYS 


167 


mitting  capacities  of  pulleys.  In  this  paper,  Prof.  Sawdon  gave 
a  table  of  relative  capacities  based  on  the  same  arc  of  contact 
and  the  same  belt  tensions,  which  may  prove  useful  in  the  solution 
of  belt  problems.  The  data  given  in  Table  39  were  derived  from 
this  paper.  In  using  the  table  it  should  be  kept  in  mind  that 
the  figures  are  relative  and,  strictly  speaking,  apply  only  to  the 
conditions  of  operation  prevailing  during  the  tests.  However, 
the  results  may  be  used  tentatively  until  further  data  per- 
taining to  this  subject  are  available. 

131.  Proportions  of  Pulleys. — (a)  Arms. — It  is  very  seldom 
that  a  designer  is  called  upon  to  design  cast-iron  pulleys  except 


FIG.  66. 


for  an  occasional  special  purpose,  and  for  that  reason  it  is  best  to 
leave  the  general  design  of  standard  pulleys  to  the  pulley  manu- 

TABLE  40. — PROPORTIONS  OF    EXTRA-HEAVY  CAST-IRON  PULLEYS 


Dimensions 


i 

2 

3 

4 

5 

6 

12 

0.38 

% 

% 

We 

m 

4 

15 

0.40 

H 

1H2 

1% 

1^6 

4K 

18 

0.42 

% 

iHe 

1% 

1% 

4^ 

24 

0.46 

1^2 

IK 

2 

2^6 

4M 

30 

0.50 

1%2 

iKe 

2%. 

2% 

7 

36 

0.54 

IK 

1% 

2^6 

3^ 

7 

42 

0.58 

1^6 

1% 

3H 

3^6 

8 

48 

0.62 

1% 

1% 

3^6 

m 

8 

54 

0.66 

1H 

2 

4^ 

5Ke 

93^ 

60 

0.70 

1^2 

2^6 

4H 

5% 

9M 

168  PROPORTIONS  OF  PULLEYS  [CHAP.  VII 

facturer.  In  Fig.  66  is  represented  an  ordinary  cast-iron  pulley, 
and  the  proportions  of  various  sizes  of  extra-heavy  double-belt 
pulleys  given  in  Table  40  may  serve  as  a  guide  in  the  design  of 
special  pulleys. 

A  series  of  tests  made  on  various  kinds  of  pulleys  by  Prof. 
C.  H.  Benjamin,  the  results  of  which  were  published  in  the 
American  Machinist  of  Sept.  22,  1898,  proved  rather  conclusively 
that  the  rim  of  a  pulley  does  not  distribute  the  torsional  moment 
equally  over  the  arms  as  is  so  frequently  assumed.  In  every 
test  made,  the  two  arms  nearest  the  tight  side  of  the  belt  gave 
way  first  and  in  almost  all  cases  rupture  of  the  arm  occurred  at 
the  hub.  As  a  result  of  these  tests,  Prof.  Benjamin  suggests  that 
the  hub  end  of  the  arm  should  be  made  strong  enough  so  that  it 
is  capable  of  resisting  a  bending  moment  equivalent  to 

M  =  2(T1-Tj,  (195) 

IV 

in  which 

D  =  diameter  of  the  pulley  in  inches. 
n  —  the  number  of  arms. 

This  means  that  one-half  of  the  arms  are  considered  as  effect- 
ive. The  dimensions  of  the  arm  at  the  rim  should  be  made  such 
that  the  sectional  modulus  is  only  one-half  of  that  at  the  hub. 

The  various  manufacturers  differ  as  to  the  number  of  arms  to 
be  used  with  the  different  sizes  of  pulleys,  but  the  following 
suggestions  may  be  found  useful: 

Use  webs  for  pulleys  having  a  diameter  of  6  inches  or  less. 

Use  4  arms  for  pulleys  having  a  diameter  ranging  from  7  to 
18  inches. 

Use  6  arms  for  pulleys  having  a  diameter  ranging  from  18  to 
60  inches. 

Use  8  arms  for  pulleys  having  a  diameter  ranging  from  60  to 
96  inches. 

When  the  face  of  a  pulley  is  wide,  a  double  set  of  arms  should 
always  be  provided. 

The  working  stress  to  be  used  in  calculating  the  dimensions  of 
the  arms  by  means  of  (195)  varies  within  very  wide  limits.  An 
investigation  of  the  arms  of  pulleys  having  a  diameter  of  from 
12  to  96  inches  and  a  face  of  4  to  12  inches  gave  stresses  varying 
from  200  to  1,500  pounds  per  square  inch.  The  latter  stress  is 
obtained  in  the  smaller  pulleys  and  the  former  with  the  larger 
diameters. 


ART.  132]  TIGHT  AND  LOOSE  PULLEYS  169 

(6)  Rim. — According  to  Mr.  C.  G.  Earth,  the  face  of  the 
pulley  should  be  considerably  wider  than  the  belt  that  is  to  run 
on  it,  and  in  order  to  establish  uniform  proportions,  he  proposed 
the  following  formulas : 

/=  1^6  + I  inch.  (196) 

/=  1^6 +  J  inch.  (197) 

Formula  (196)  is  the  one  that  should  be  used  wherever  possible, 
but  occasionally  due  to  certain  restrictions  as  to  available  space, 
(197)  may  have  to  be  used.  In  connection  with  these  formulas, 
Mr.  Barth  recommends  that  the  height  of  the  crown  should  be 

determined  by  the  formula 

m 
c  =  -L  (198) 

For  proportions  of  the  thickness  of  the  rim,  the  data  given  in 
Table  40  may  be  of  service. 

132.  Tight  and  Loose  Pulleys. — In  his  consulting  work,  Mr. 
Barth  has  found  the  need  of  well-designed  tight  and  loose  pulleys. 
After  a  thorough  study  of  the  conditions  under  which  such  pul- 
leys must  operate,  he  developed  the  design  shown  in  Fig.  67. 
Furthermore,  he  standardized  the  design,  and  the  formulas  be- 
low give  well-proportioned  sleeves  and  pulleys  for  shaft  diameters 
from  1%  to  4  inches,  inclusive.  The  face  and  height  of  crown 
for  these  pulleys  are  based  on  formulas  (196)  to  (198)  inclusive. 
The  formulas  giving  the  proportions  of  the  pulley  hub  and  sleeve 
a  are  based  on  the  diameter  d  of  the  shaft. 

di  =  1.5  d  +  1.5  inches 

d2  =  1.5  d  +  1  inch 

d3  =  1.375  d  +  0.75  inch 

di  =  1.25  d  +  0.25  inch  (199) 

€  =  A  +  0.125  inch 
lo 

m  =  -  -f  0.75  inch 
o 

The  formulas  listed  below  give  proportions  of  the  loose  pulley 
rim,  and  are  based  upon  the  width  of  the  belt  running  on  the 
pulleys.  The  belt  width  as  given  by  Mr.  Barth  varied  from  2 
to  6  inches,  inclusive. 


170 


TIGHT  AND  LOOSE  PULLEYS  [CHAP.  VII 


-ll  5  +  f  inch 


L  = 


b         A  . 


3    • 


(200) 


(b) 
FIQ.  67. 


ART.  133] 


V  BELTING 


171 


The  common  tight  and  loose  pulleys  that  are  used  in  the 
majority  of  installations  differ  considerably  from  the  design 
discussed  above  in  that  both  pulleys  are  generally  made  alike, 
and  in  many  cases  neither  pulley  is  crowned. 


V   BELTING 

133.  Types  of  V  Belts.— As  stated  in  the  first  part  of  this 
chapter,  V  belts  are  used  when  it  is  desired  to  transmit  light 
power;  for  example,  in  driving  the  cooling  fan  and  generator  on 


r_r? 


(a) 


(b) 


FIG.  68. 

automobiles,  and  transmission  drives  on  motorcycles.  It  is 
also  used  for  belting  electric  motors  to  pumps  and  ventilating 
fans,  when  the  distances  between  the  shafts  are  short.  Several 
forms  of  V  belting  are  shown  in  Fig.  68. 

(a)  Block  type. — The  construction  used  in  the  block  type  of 
V  belt  is  shown  in  Fig.  68 (a).  It  consists  of  a  plain  high-grade 
and  very  pliable  leather  belt  to  which  are  cemented  and  riveted 


172  V  BELTING  [CHAP.  VII 

equally  spaced  V  blocks,  also  made  of  leather.  For  light  loads, 
a  single  belt  is  used;  and  for  heavy  service,  a  wide  belt  is  fitted 
with  several  rows  of  V  blocks.  The  angle  adopted  in  this  design 
is  28  degrees,  and  according  to  the  manufacturers  of  this  belt, 
the  maximum  speed  should  not  exceed  3,000  feet  per  minute. 
The  belt  shown  in  Fig.  68 (a)  is  also  used  successfully  on  high 
pulley  ratios,  though  the  best  results  are  obtained  if  the  ratio 
does  not  exceed  6  or  7  to  1.  In  addition  to  giving  good  service 
on  high-ratio  pulleys,  the  block  type  of  V  belt  also  works  suc- 
cessfully on  pulleys  located  close  together.  The  following  recom- 
mendations were  furnished  by  the  Graton  and  Knight  Mfg. 
Co.: 

1.  For  a  2,  3,  or  4  to  1  ratio,  the  minimum  center  distance 
equals  the  diameter  of  the  larger  pulley  plus  twice  the  diameter 
of  the  smaller  one. 

2.  For  a  5,  6,  or  7  to  1  ratio,  the  minimum  center  distance 
equals  the  diameter  of  the  larger  pulley  plus  three  times  the 
diameter  of  the  smaller  one. 

3.  For  a  8,  9,  or  10  to  1  ratio,  the  minimum  center  distance 
equals  the  diameter  of  the  larger  pulley  plus  four  times  the 
diameter  of  the  smaller. 

(b)  Chain  type. — The  construction  shown  in  Fig.  68(6)  is  of 
the  chain  type,  and  consists  of  double  links  made  of  oak-tanned 
sole  leather  connected  together  by  central  links  c  made  of  steel. 
The  steel  links  are  fitted  with  short  pins  d  to  which  the  leather 
links  are  attached.  To  add  strength  to  the  belt  as  well  as  to 
afford  a  fair  bearing  for  the  pins  d,  vulcanized  fiber  links  6  are 
used  between  the  leather  and  steel  links.  An  ordinary  wood 
screw  clamps  the  two  sets  of  double  links  together,  as  illustrated 
in  the  figure.  All  the  driving  is  done  by  the  leather  links,  and 
the  angle  used  is  28  degrees. 

Another  construction  of  the  chain  type  V  belt  made  entirely 
of  steel,  except  the  part  coming  into  contact  with  the  pulley,  is 
shown  in  Fig.  68  (c) .  The  material  used  for  lining  the  steel  driving 
members  is  not  leather  but  a  specially  treated  asbestos  fabric. 

134.  Force  Analysis  of  V  Belting. — To  determine  the  relation 
existing  between  the  tight  and  loose  tensions  in  a  V-belt  power 
transmission,  we  may  follow  the  method  given  in  Art.  122. 

Let  w  =  weight  per  foot  of  belt. 
2  j(3  =  total  angle  of  the  V  groove. 
C,  Vj  jj,  and  6  same  meaning  as  in  Art.  122. 


ART.  134] 


V  BELTING 


173 


Referring  to  Fig.  69  and  taking  the  summation  of  the  horizontal 
and  vertical  components,  respectively,  of  all  forces  acting  upon  a 
small  portion  of  the  belt,  we  get 


AT  cos  ~  - 
A0 


0 


(2  T  +  AT7)  sin  —  -  2  N  sin  0  -  C  =  0 


(201) 
(202) 


The  magnitude  of  the  centrifugal  force  C  in  this  case  is  given 
by  the  following  equation: 

C  =  ^^  (203) 

Eliminating  N  in  (201)  and  (202)  and  taking  the  limits  of  the 
resultant  expression,  we  finally  get 

dT 


T  _  wv^      sin  ft 


dd 


(204) 


FIG.  69. 

Integrating  (204)  between  the  proper  limits  for  T  and  6,  we 
obtain 


1 1 


esin/3 


(205) 


The  net  driving  tension  of  the  belt  is 


gsin/3    —    1 


sin./S 


(206) 


To  determine  the  horse  power  transmitted,  substitute  the 
magnitude  of  the  net  driving  tension  obtained  from  (206)  in 
(193). 


174  REFERENCES  [CHAP.  VII 

References 

Die  Maschinen  Elemente,  by  C.  BACH. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Leather  Belting,  by  R.  T.  KENT. 

Experiments  on  the  Transmission  of  Power  by  Belting,  Trans.  A.  S.  M.  E., 
vol.  7,  p.  549. 

Belt  Creep,  Trans.  A.  S.  M.  E.,  vol.  26,  p.  584. 

The  Transmission  of  Power  by  Leather  Belting,  Trans.  A.  S.  M.  E.,  vol.  31, 
p.  29. 

The  Effect  of  Relative  Humidity  on  an  Oak-tanned  Leather  Belt,  Trans. 
A.  S.  M.  E.,  vol.  37,  p.  129. 

Tensile  Tests  of  Belts  and  Splices,  Amer.  Mack.,  Oct.  10,  1912. 

Belt  Driving,  The  Engineer  (London),  Apr.  23  and  30,  1915. 

The  Design  of  Tandem  Belt  Drives,  Amer.  Mach.,  Apr.  1,  1915. 

Theory  of  Steel  Belting,  Zeitschrift  des  Vereins  Deutscher  Ingenieure,  Oct. 
21,  1911. 

Transmission  of  Power  by  Means  of  Steel  Belting,  Dinglers,  Sept.  2  and  9, 
1911. 

The  Practicability  of  Steel  Belting,  Amer.  Mach.,  Nov.  21,  1912. 


CHAPTER  VIII 
MANILA  ROPE  TRANSMISSION 

Ropes  used  in  engineering  operations  are  made  of  a  fibrous 
material  such  as  manila,  hemp  and  cotton,  or  of  iron  and  steel. 
As  to  the  kind  of  service,  ropes  may  be  classed  as  follows:  (a) 
those  used  for  the  hoisting  and  transporting  of  loads;  (6)  those 
used  for  the  transmission  of  power. 

FIBROUS  HOISTING  ROPES 

135.  Manila  Hoisting  Rope. — Manila  rope  is  manufactured 
from  the  fiber  of  the  abaca  plant,  which  is  found  only  in  the 
Philippine  Islands.     It  has  a  very  high  tensile  strength,  tests 
made  at  the  Watertown  Arsenal  showing  that  it  exceeds  50,000 
pounds  per  square  inch.     In  making  the  rope,  the  fibers  are 
twisted  right-handed  into  yarns;  these  yarns  are  then  twisted 
in  the  opposite  direction  forming  the  strands,  and  to  form  the 
finished  rope  a  number  of  strands  are  twisted  together,  again  in 
the  right-hand  direction. 

Practically  all  manila  rope  used  for  hoisting  purposes  has  four 
strands  except  the  sizes  below  %  inch,  which  are  made  with  three 
strands.  For  drum  hoists  using  manila  ropes,  the  maximum 
speed  attained  under  load  seldom  exceeds  1,000  feet  per  minute, 
generally  being  nearer  300  feet  per  minute.  The  permissible 
working  loads  of  the  various  sizes  of  manila  ropes  used  for  hoist- 
ing service  are  given  in  Table  41. 

136.  Sheave  Diameters. — A  rope  in  passing  over  sheaves  is 
subjected  to  a  considerable  amount  of  internal  wear,  due  to  the 
fibers  sliding  upon  each  other.     The  smaller  the  diameter  of  the 
sheave  the  greater  this  sliding  action  becomes ;  hence  to  decrease 
the  wear,  large  sheaves  should  be  used.     In  addition  to  the 
internal  wear  there  is  also  wear  on  the  outside  of  the  rope  due 
to  the  friction  between  it  and  the  sides  of  the  grooves  of  the 
sheave.     It  is  evident,  therefore,  that  the  grooves  should  be 
finished  very  smooth.     Again,  the  arrangement  of  the  various 
elements  that  make  up  the  hoisting  apparatus  may  be  such  that 
an  excessive  number  of  bends  is  introduced,  thus  increasing  the 
wear. 

175 


176 


MANILA  ROPE 
TABLE  41. — MANILA  ROPE 


[CHAP.  VIII 


Diameter 
in 
inches 

For  hoisting 

For  Transmission 

Weight 
per 
foot 

Ultimate 
strength 

Mini- 
mum 
sheave 
diam. 

Weight 
foot 

Ultimate 
strength 

Maximum 
allowable 
tension 

Mini- 
mum 
sheave 
diam. 

V* 

0.018 

620 

%6 

0.024 

1,000 

H 

0.037 

1,275 

KG 

0.055 

1,875 

X 

0.075 

2,400 

%6 

0.104 

3,300 

% 

0.133 

4,000 

y± 

0.16 

4,700 

0.21 

3,950 

112 

28 

V* 

0.23 

6,500 

0.27 

5,400 

153 

32 

i 

0.27 

7,500 

8 

0.36 

7,000 

200 

36 

IK 

0.36 

10,500 

9 

0.45 

8,900 

253 

40 

m 

0.42 

12,500 

10 

0.56 

10,900 

312 

46 

1% 

0.55 

15,400 

11 

0.68 

13,200 

378 

50 

IH 

0.61 

17,000 

12 

0.80 

15,700 

450 

54 

1% 

0.75 

20,000 

13 

0.92 

18,500 

528 

60 

1% 

0.93 

25,000 

14 

1.08 

21,400 

612 

64 

2 

1.09 

30,000 

1.40 

28,000 

800 

72 

2K 

1.5 

37,000 

1.80 

35,400 

1,012 

82 

2K 

1.71 

43,000 

2.20 

43,700 

1,250 

90 

Experience  has  shown  that  manila  ropes  give  good  service  and 
will  last  a  reasonable  length  of  time  in  hoisting  operations  when 
the  sheaves  for  the  various  sizes  of  ropes  are  made  according  to 
the  diameters  given  in  Table  41. 

137.  Stresses  in  Hoisting  Ropes. — In  hoisting  operations  ropes 
are  wound  upon  drums,  and  sheaves  are  used  for  changing  the 
direction  of  the  rope.  In  passing  over  sheaves  or  onto  drums, 
the  rigidity  of  the  rope  offers  a  resistance  to  bending  which  must 
be  overcome  by  the  effort  applied  to  the  pulling  side  of  the  rope. 
To  determine  the  relation  that  exists  between  the  effort  P  and 
the  resistance  Q  for  a  rope  running  over  a  guide  sheave,  the 
following  method  may  be  used : 

Let  D  =  pitch  diameter  of  the  sheave. 
d  =  diameter  of  the  sheave  pin. 
M  =  coefficient  of  journal  friction, 
r?  =  efficiency. 


ART.  137] 


STRESSES  IN  HOISTING  ROPE 


177 


On  the  running-on  side  of  the  sheave  shown  in  Fig.  70,  the 
outer  fibers,  due  to  the  bending  of  the  rope,  are  in  tension  while 
the  inner  fibers  are  in  compression.  These  tensile  and  com- 
pressive  stresses  when  combined  with  the  tension  distributed 
uniformly  over  the  section  will  produce  a  resultant  which  has  its 


point  of  application  to  the  left  of  the  center  line  of  the  rope, 
a  distance  designated  by  the  symbol  s.  The  resultant  must  be 
equal  to  Q,  from  which  it  follows  that  rope  stiffness  may  be  con- 
sidered as  having  the  same  effect  as  increasing  the  lever  arm  of 
the  resistance  Q. 

By  applying  the  same  line  of  reasoning  to  the  running-off  side, 
it  may  be  shown  that  the  rigidity  of  the  rope  has  the  effect  of 
decreasing  the  lever  arm  of  the  effort  P  by  an  amount  which  may 
be  taken  as  approximately  equal  to  s.  Introducing  friction  at 
the  sheave  pin  and  taking  moments  about  the  line  of  action  of 
the  resultant  pressure  upon  this  pin,  we  obtain 


(207) 


Since  the  efficiency  of  a  mechanism  is  defined  as  the  ratio  of 
the  useful  work  done  to  the  total  work  put  in,  it  is  evident  that 
in  the  case  of  the  ordinary  rope  guide  sheave 


« 


(208) 


178 


BLOCK  AND  TACKLE 


[CHAP.  VIII 


138.  Analysis  of  Hoisting  Tackle. — Analyses  of  systems  of 
hoisting  tackle  or  so-called  pulley  blocks  are  readily  made  with 
the  aid  of  the  principle  discussed  in  the  preceding  article.  The 

application  of  this  prin- 
ciple will  be  shown  by 
an  example. 

Common  block  and 
tackle. — The  common 
block  and  tackle  con- 
sists of  two  pulley 
blocks,  each  block  hav- 
ing a  series  of  sheaves 
mounted  side  by  side  on 
the  same  axle  or  pin. 
The  number  of  sheaves 
varies  in  ordinary  hoist- 
ing operations  from  two 
to  four,  but  when  used  in  connection  with  wire  rope  on  hydraulic 
elevators  or  on  cranes  these  numbers  are  exceeded.  For  con- 
venience of  analysis,  we  may  assume  the  sheaves  of  each  block 
to  be  placed  on  separate  pins  as  shown  in  Fig.  71.  Beginning 
with  the  end  of  the  rope  fastened  to  the  upper  block,  let  the  suc- 
cessive tensions  in  the  parts  of  the  rope  supporting  the  load  Q, 
be  denoted  by  T\,  T2,  etc. ;.  then 


(209) 


FIG.  71. 


-/  4    —    ^    J-  l')        J-  5    —    ^    J-  1)   J-  6    —    ^    J-  1 

P  =  C6Tl 

Q  =  TG  +  T,  +  T,  +  T3  +  T2  +  Ti 


(210) 


Substituting  the  value  of   T\  from  (209)  in  (210),  we  obtain 

C  - 


P  =  C6 
Without  friction,  the  effort  required  to  raise  the  load  Q  is 

Po=Q 

Hence  the  efficiency  for  the  tackle  shown  in  Fig.  71  is 

C6  -  1 

77  ~  6C6(C^T) 


(211) 


(212) 


(213) 


AET.  139] 


DATA  ON  HOISTING  TACKLE 


179 


In  general  when  the  block  and  tackle  has  n  sheaves  and  n 
lines  supporting  the  load  Q,  we  get  as  the  general  expression  for 
the  effort 


p  = 


and  for  the  efficiency 


(215) 


139.  Experimental  Data  on  Hoisting  Tackle.  —  Experimental 
data  on  hoisting  tackle  reefed  with  manila  rope  are  meager,  so 
in  order  to  obtain  some  information  as  to  the  efficiency  of  such 

TABLE  42.  —  HOISTING  TACKLE  REEFED  WITH  MANILA  ROPE 


Size  of 
rope 

Block  and  tackle  data 

Ratio 
Q/P 

Value  of  C 

Sheave 
diam. 

Pin 

diam. 

No.  of 
sheaves 

No.  of 
lines 

Test 

Mean 

*     1 

2 

1.92 

1.087 

2 

3 

2.68 

1.125 

IK 

m 

% 

'    3 

4 

4 
5 

3.37 
3.95 

1.127 
1.135 

1.13 

5 

6 

4.48 

1.13 

6 

7 

4.92 

1.14 

1 

2 

1.91 

1.098 

2 

3 

2.67 

1.125 

3 

4 

3.36 

1.134 

1H 

9% 

1 

4 
5 

5 
6 

3.93 
4.45 

1.14 
1.141 

1.14 

6 

7 

4.89 

1.143 

7 

8 

5.28 

.143 

8 

9 

5.61 

.143 

2 

3 

2.64 

.136 

3 

4 

3.30 

.142 

4 

5 

3.84 

.155 

m 

10% 

1H 

5 

6 

4.33 

.155 

1.15 

6 

7 

4.72 

.158 

7 

8 

5.08 

.162 

8 

9 

5.37 

.16 

4 

5 

3.87 

.15 

5 

6 

4.37 

.15 

2 

13 

IH 

6 

7 

4.78 

.153 

1.15 

7 

8 

5.14 

.152 

o 

9 

5.45 

.153 

180  MULTIPLE  SYSTEM  [CHAP.  VIII 

apparatus,  the  American  Bridge  Co.  made  an  extended  series 
of  tests  at  the  Pencoyd  plant.  These  tests  were  made  with 
standard  types  of  manila  and  wire  rope  blocks,  and  an  attempt 
was  made  to  reproduce  as  nearly  as  possible  actual  conditions 
under  which  such  apparatus  is  used  in  practice.  The  results  of 
these  tests  were  reported  by  S.  P.  Mitchell  in  a  paper  entitled 
" Tests  on  the  Efficiency  of  Hoisting  Tackle"  and  were  presented 
before  the  American  Society  of  Civil  Engineers  in  September, 
1903.  That  part  of  the  data  pertaining  to  manila  ropes  is 
given  in  Table  42.  In  the  last  two  columns  of  this  table  are 
given  the  values  of  C  as  determined  by  means  of  equation  (214). 

FIBROUS  TRANSMISSION  ROPE 

Leather  belting,  while  excellent  for  transmitting  power  for 
short  distances  under  cover,  is  not  suitable  for  transmitting 
power  to  long  distances  out  of  doors,  and  for  this  class  of  service, 
manila  and  cotton  ropes  are  used.  Cotton  rope,  however,  is  not 
used  to  any  extent  in  this  country.  The  construction  of  the 
manila  rope  used  for  the  transmission  of  power  is  similar  to  that 
discussed  in  Art.  135. 

The  transmission  of  power  by  means  of  manila  rope  gives 
satisfactory  results  for  distances  between  shafts  as  great  as  one 
hundred  and  seventy-five  feet  without  the  use  of  carrying  pulleys, 
while  with  the  carriers,  the  distance  may  be  increased  almost  in- 
definitely. Manila  rope  is  also  well-adapted  to  short  distances. 
By  the  use  of  properly  located  guide  pulleys  power  may  be  trans- 
mitted from  one  shaft  to  another,  no  matter  what  the  relative 
positions  of  the  shaft.  There  are  two  systems  of  rope  driving  in 
use,  and  each  has  its  advocates.  The  two  systems  are  com- 
monly called  the  Multiple  or  English  System  and  the  Continuous 
or  American  System. 

140.  Multiple  System. — The  multiple  system,  which  is  the 
simpler  of  the  two,  uses  separate  ropes  each  spliced  into  an 
endless  belt  and  running  in  a  separate  groove  on  each  sheave 
wheel;  thus  each  rope  is  absolutely  independent  of  any  other  and 
carries  its  proportion  of  the  load.  The  last  statement  is  only 
true  if  the  ropes  are  spliced  carefully  and  the  initial  tension  in 
each  rope  is  made  the  same.  The  multiple  system  may  be  used 
for  heavy  loads  and  is  recommended  where  the  drive  is  protected 
from  the  weather  and  when  the  shafts  are  parallel  or  approxi- 


ART.  141]  CONTINUOUS  SYSTEM  181 

mately  so,  as  in  installations  where  the  power  from  a  prime  mover 
has  to  be  distributed  to  the  several  floors  of  a  building.  This 
system  also  finds  favor  for  rolling  mill  service,  in  which  service 
it  is  common  practice  to  install  several  more  ropes  than  are. ab- 
solutely necessary  to  transmit  the  power  so  that  the  mill  need 
not  be  closed  down  even  if  several  of  the  ropes  should  fail  or 
jump  off. 

The  advantages  possessed  by  the  multiple  system  are  as 
follows : 

1.  It  is  practically  secure  against  breakdowns,  and  if  a  rope 
should   break  it  may  be  removed  and  replaced  at  some  con- 
venient time. 

2.  The  power  transmitted  may  be  increased  by  adding  extra 
ropes. 

3.  Power  may  be  more  easily  transmitted  to  the  different 
floors  of  an  establishment. 

4.  The  life  of  a  rope  is  greater  than  in  the  continuous  system, 
since  it  always  bends  in  the  same  direction. 

5.  It  is  cheaper  to  install. 

Among  the  disadvantages  are  the  following: 

1.  It  has  more  slippage  than  the  continuous  system. 

2.  It  is  not  well-adapted  to  quarter  turn  drives  nor  where  the 
shafts  are  at  an  angle  with  each  other. 

141.  Continuous  System. — In  the  continuous  system  one 
continuous  rope  passes  around  the  driving  and  driven  sheaves 
several  times,  in  addition  to  making  one  loop  about  a  tension 
pulley  located  on  a  traveling  carriage.  Since  a  single  rope  is 
used,  it  is  evident  that  some  device  is  required  that  will  lead  the 
rope  from  the  outside  groove  of  the  driving  sheave  to  the  oppo- 
site outside  groove  of  the  driven  sheave.  This  device  is  the  ten- 
sion pulley.  Other  functions  of  this  traveling  tension  pulley  are 
to  maintain  continually  a  definite  uniform  tension  in  the  rope,  and 
to  take  care  of  the  slack  due  to  the  stretching  of  the  rope.  In  Fig. 
72,  is  shown  one  way  of  taking  care  of  the  slack  by  means  of  a 
tension  carriage. 

The  continuous  system  is  well-adapted  to  vertical  and  quarter 
turn  drives,  and  to  installations  having  shafts  that  are  at  an 
angle  to  each  other.  It  also  gives  better  service  in  places  where 
the  rope  is  exposed  to  the  weather.  The  following  are  some  of 
the  disadvantages: 


182  MANILA  TRANSMISSION  ROPE  [CHAP.  VIII 

1.  A  break  in  the  rope  shuts  down  the  whole  plant  until  the 
rope  is  spliced  and  again  placed  on  the  sheaves. 

2.  All  of  the  ropes  are  not  subjected  to  the  same  tension;  that 
is,  the  rope  leading  from  the  tension  carriage  has  a  greater  ten- 
sion than  the  center  ropes. 

142.  Manila  Transmission  Rope. — For  the  transmission  of 
power,  the  four  or  six-strand  ropes  are  used  on  all  sizes  above 
%-inch.  For  the  %-inch  size,  which  is  the  smallest  transmission 
rope  made,  the  three-strand  type  gives  good  service.  The  four- 
and  six-strand  ropes  of  both  hoisting  and  transmission  types 
have  the  strands  laid  around  a  core  which  has  been  treated  with 
a  lubricant.  A  lubricant  is  used  also  on  the  inner  yarns  of  each 


FIG.  72. 

strand,  thus  insuring  proper  lubrication  of  the  rope.  For  trans- 
mission purposes  experience  shows  that  the  best  results  are  ob- 
tained when  the  speed  of  the  rope  is  approximately  4,500  feet 
per  minute.  Higher  speeds  are  used,  but  the  life  of  the  rope  is 
decreased  due  to  excessive  wear. 

143.  Sheaves. — The  diameter  of  a  sheave,  used  in  the  trans- 
mission of  power  by  means  of  manila  ropes,  should  be  made 
forty  times  the  diameter  of  the  rope  when  space  and  speeds 
permit.  Sometimes  it  is  necessary,  due  to  constructive  reasons, 
to  make  the  diameter  less  than  that  called  for  by  the  above  rule. 
This  reduction  of -the  diameter  decreases  the  life  of  the  rope  very 
materially  and  it  is  well  to  keep  the  minimum  diameter  above 
thirty-six  times  the  diameter  of  the  rope. 

Form  of  groove. — The  forms  of  the  grooves  used  in  the  two 
systems  of  transmission  discussed  in  the  preceding  articles  differ 
somewhat,  although  in  the  angle  used  by  some  of  the  manu- 


ART.  143] 


SHEAVES 


183 


facturers,  they  are  similar.  Experience  seems  to  show  that  an 
angle  of  45  degrees  gives  the  best  results  for  both  systems. 
However,  there  are  one  or  two  manufacturers  of  rope  transmis- 
sions that  recommend  an  angle  of  60  degrees.  In  Fig.  73  are 
shown  the  forms  of  grooves  recommended  for  the  continuous 
system,  (a)  and  (6)  being  used  for  the  driver  as  well  as  the  driven, 
and  (c)  for  the  idler  sheaves.  As  illustrated  in  the  figure,  the 
grooves  are  not  made  deep  since  the  rope  is  kept  taut  in  order  to 


(c) 


FIG.  73. 


decrease  the  tendency  for  it  to  jump  out.  The  type  of  groove 
shown  in  Fig.  73 (a)  is  used  by  the  Allis-Chalmers  Co.;  for 
proportions  thereof  consult  Table  43.  For  proportions  of  the 
form  of  groove  used  by  the  Dodge  Mfg.  Co.  illustrated  in  Fig. 
73(6)  consult  Table  43. 

The  form  of  groove  commonly  used  in  the  multiple  system,  and 
occasionally  in  the  continuous  system,  is  shown  in  Fig.  73  (d), 
and  in  Table  44  are  given  the  proportions  of  this  groove  for  the 
various  sizes  of  transmission  ropes.  The  form  of  the  groove 
used  on  idlers  with  the  multiple  system  is  deeper  than  that  shown 
in  Fig.  73 (c),  but  in  other  details  it  is  about  the  same. 


184 


DIMENSIONS  OF  GROOVES 


[CHAP.  VIII 


TABLE  43. — DIMENSIONS  OF  GROOVES  FOR  MANILA  ROPE  SHEAVES 
All  dimensions  in  inches 


Allis-Chalmers  standard 

Dodge  Mfg.  Co.  standard 

Size  of 

rope 

Pitch 

1 

2 

3 

4 

5 

6 

Pitch 

1 

2 

3 

4 

K 

IK 

1 

%6 

K 

K 

1% 

1M 

^ 

%6 

1 

1>£ 

1%6 

l^6 

1 

^i 

H 

Me 

1>£ 

IK 

1He 

^^ 

IK 

1% 

1% 

K 

1Me 

K 

IK 

IK 

IK 

IK 

IKe 

1Me 

y* 

>i 

12£ 

IK 

1%6 

K 

IK 

IK 

1^ 

H 

X}Hl6 

IK 

2 

2-H6 

1« 

lIKe 

\y% 

H 

>^ 

2 

IK 

1 

1Me 

IK 

2H 

2XM6 

!1Ke 

1%6 

1% 

H 

y± 

2M 

2 

IK 

i^6 

\  ax 

2 

m 

2K 

1MK. 

1« 

IN 

1 

K 

w 

2K 

IK 

1H. 

/ 

TABLE  44. — DIMENSIONS  OF  GROOVES  FOR  MANILA  ROPE  SHEAVES 
All  dimensions  in  inches 


Size  of 
rope 


Pitch 


Engineers  standard 


/3£ 

7^: 


2% 


2^6 
3% 
3K 
4 

3^6 

3%6 


lMa 

IKe 


IK 

2 


1 

IK 
1K« 


KG 
K 


•K 


H 

H 


144.  Relation  between  Tight  and  Loose  Tensions. — In  order 
to  calculate  the  horse  power  transmitted  by  a  manila  rope  at  a 
given  speed,  it  is  necessary  to  know  the  net  tension  on  the  ropes, 
and  to  get  this  we  must  determine  the  relation  existing  between 
the  tight  and  loose  tensions.  Due  to  the  wedging  action  of  the 
rope  in  the  groove  of  the  sheave,  the  friction  between  the  sheave 
and  the  rope  is  considerably  greater  than  for  the  case  of  plain 
belting.  The  ratio  between  the  tensions  may  be  derived  by  the 
same  method  as  that  given  in  Art.  134.  Using  the  same  notation 
as  in  the  discussion  of  the  V  belting,  and  considering  a  short 
length  of  the  rope  having  an  arc  of  contact  subtending  the  angle 


ART.  144]  RATIO  OF  ROPE  TENSIONS  185 

Ad  at  the  center  of  the  sheave,  we  get  for  the  summation  of  the 
horizontal  and  vertical  components,  respectively 

AT  cos  ~  -  2  »N  =  0  (216) 


(2  T  +  AT7)  sin        -  2  N  sin  0  -  C  =  0  (217) 

z 

Proceeding  as  in  Art.  134,  we  finally  obtain 

T    -  —          »e 

---  ^  =  <F*  =  &"  (218) 

z 


T 

With  the  usual  conditions  under  which  manila  ropes  run,  the 
coefficient  of  friction  /z  may  be  assumed  as  0.12,  and  the  angle 
2  ]8  as  given  in  Art.  143  may  be  either  45  or  60  degrees.  Using 

these  coefficients,  the  values  of  -  —  ~  are  as  follows: 

sin  p 

For  45-degree  groove,  /  =  0.314. 
For  60-degree  groove,  //  =  0.24. 

Horse  power.  —  As  in  the  case  of  belt  transmission,  the  horse 
power  is  given  by  the  formula 

!  -  r2)  (219) 

From  (218),  the  net  driving  tension  is  given  by  the  following 
expression  : 

r,  -  r.  -  [Tl  -  ^  [^]  (220) 

Therefore 

v  r  wv2-}  re*'e  -  li  .      . 

=          Tl  "  ~-- 


It  is  important  to  note  that  there  is  a  rope  speed  that  makes 
the  horse  power  transmitted  a  maximum,  and  beyond  which 
the  horse  power  decreases.  An  expression  for  the  speed  corre- 
sponding to  the  maximum  horse  power  may  be  determined  by 
equating  the  first  derivative  of  H  with  respect  to  v  to  zero,  and 
solving  for  v.  Thus  from  (221) 


dH 

—7- 

dv 


T 

I 


186  ANALYSIS  OF  A  TRANSMISSION 

whence  for  maximum  H 


[CHAP.  VIII 


(222) 


The  general  form  of  the  curve  expressing  the  relation  between 
the  horse  power  and  the  rope  speed  is  shown  in  Fig.  74.  The 
full  line  applies  to  a  IJ^-inch  rope  running  on  a  sheave  having  a 
45-degree  groove,  while  the  broken  line  applies  to  the  same  size 
of  rope  using  a  60-degree  groove.  In  plotting  these  graphs,  it 


01 


o 


oi 


o 


\ 


20  40  60  60  100 

Speed     of    Rope-  ft.  per  sec. 
FIG.  74. 


120 


140 


was  assumed  that  the  coefficient  of  friction  was  the  same  for  both 
cases. 

145.  Force  Analysis  of  a  Manila  Rope  Transmission. — As 

stated  in  Art.  141,  one  of  the  functions  of  the  tension  carriage  is 
to  produce  a  uniform  tension  in  the  ropes,  but  the  following 
analysis  will  disclose  that  such  a  condition  is  not  realized  in  the 
continuous  system.  In  Fig.  72  is  shown  diagrammatically  what 
is  known  as  the  American  Open  Drive.  It  should  be  noticed  that 
the  tension  carriag3  is  located  just  off  the  driving  sheave.  From 


ART.  145] 


ANALYSIS  OF  A  TRANSMISSION 


187 


the  discussion  in  Art.   137  and  144,  we  readily  arrive  at  the 
following  relations : 

(223) 


T,  ' 

T2 

Tt* 

T\  __ 

TT\  1^  \  (224) 


TI  ' 

The  total  net  tension  on  the  driving  sheave  is  the  difference  of 
the  sum  of  the  tensions  on  the  tight  and  slack  sides,  or 


T  = 


(225) 


Now  combining  (224)  with  (225),  the  net  tension  T  may  be 
obtained  in  terms  of  T$  and  known  constants;  hence,  the  magni- 
tude of  T8  is  fully  determined  since  the  horse  power  transmitted 
and  the  rope  speed  are  known.  Knowing  T8,  (223)  enables  us 
to  establish  the  magnitude  of  the  tension  P. 

By  comparing  the  expressions  for  T2,  TI  and  T6  it  is  evident 
that  these  tensions  are  not  of  the  same  magnitude,  but  that  each 
successive  tension  on  the  tight  side  is  smaller  than  the  one  pre- 
ceding it.  The  same  is  true  on  the  slack  side.  To  overcome  this 
inequality  in  the  tension  of  the  various  ropes  running  over 
sheaves  of  unequal  diameter,  the  above  analysis  shows  that 
either  one  of  the  following  methods  could  be  used : 

1.  By  using  sheaves  of  different  materials,  thus  changing  the 
coefficient  of  friction  n  so  that  jui#i  =  M202. 

2.  By  using  the  same  material  for  both  sheaves,  but  changing 
the  angle  of  the  grooves  so  that  AH#I  =  M202. 

The  latter  method  is  the  more  practical  and  installations  using 
this  scheme  are  in  successful  operation.  Mr.  Spencer  Miller 
was  probably  the  first  one  to  advocate  using  different  groove 
angles  on  driving  and  driven  sheaves  of  unequal  diameters.  The 
subject  was  discussed  by  Mr.  Miller  in  a  paper  read  before  the 


188  SHEAVE  PRESSURES  [CHAP.  VIII 

American  Society  of  Civil  Engineers  in  June,  1898,  and  reported 
in  volume  39,  page  165  of  the  Transactions  of  that  society. 

146.  Sheave  Pressures.  —  The  series  of  equations  given  by 
(224)  above  enables  us  to  determine  the  approximate  pressures 
coming  upon  the  shafts  of  the  sheaves  due  to  the  rope  tensions. 
The  pressure  upon  the  shaft  of  the  driving  sheave,  assuming  the 
tight  and  slack  side  to  be  practically  parallel,  is 

Qi  =  T2  +  Tz  +  Ti  +  T,  +  T,  +  T7         (226) 
The  pressure  upon  the  shaft  of  the  driven  sheave  is 

Q2  =  Ti  +  T2  +  T,  +  T,  +  T,  +  T6         (227) 

The  pressures  upon  the  shafts  of  the  idler  sheaves  a  and  b  are 
respectively 

Qs  =  T7  +  T8,  (228) 

and 

Q4  =  T!  +  Ts  (229) 

The  horse  power  absorbed  by  the  friction  of  the  bearings  on  the 
shafts,  due  to  the  pressure  just  determined,  is  considerable  and 
may  be  estimated  by  the  following  expression  : 


Hf  =       (QlNiri  +  QzN^  +  Q*N*r*  +  Q*N^>  (230) 


in  which  N  denotes  the  number  of  revolutions  per  minute  of 
the  sheave,  r  the  radius  of  the  sheave  shaft,  and  ju3  the  coefficient 
of  journal  friction. 

147.  Sag  of  Rope.  —  In  practically  all  rope  transmissions  it  is 
important  to  determine  the  approximate  sag  of  the  ropes.  In 
arriving  at  a  formula  by  means  of  which  the  probable  sag  may 
be  estimated,  no  serious  error  is  introduced  by  assuming  that  the 
rope  hangs  in  the  form  of  a  parabola  instead  of  a  catenary.  In 
Fig.  75  is  shown  a  rope  suspended  over  two  sheaves,  the  line 
ABC  representing  approximately  the  curve  assumed  by  the  rope. 
From  the  equation  of  the  parabola  we  have 

r2        T2 

|i  =  £  (231) 

hi         n2 

Substituting  the  value  of  L2  =  L  —  LI  in  (231)  and  reducing 
the  expression  to  the  simplest  form,  we  finally  get 

(232) 


ART.  147] 

In  a  similar  manner 


SAG  OF  ROPE 


189 


(233) 


The  horizontal  tension  in  the  rope  at  the  lowest  point  B  is 

T   2  T    2 


T  •= 


2  hi        2h2 


(234) 


The  difference  in  the  tensions  at  any  two  points  of  a  rope  form- 
ing a  catenary  is  equal  to  the  difference  in  elevation  of  these 
points  multiplied  by  the  weight  per  unit  length  of  rope.  Treat- 
ing the  rope  ABC  in  Fig.  75  as  if  it  formed  a  catenary  and  applying 
the  property  just  mentioned,  the  tension  Ta  at  A  is 


Ta  =  T  + 


(235) 


FIG.  75. 


and  the  tension  at  C  is 
T,  = 


(236) 


From  (235),  it  follows  that  the  magnitude  of  the  sag  hi  is  given 
by  the  following  expression : 


-  2 


and  from  (236),  the  sag  h%  is 


-  2 


(237) 


(238) 


By  means  of  (237)  and  (238)  the  sag  of  the  ropes  on  either  the 
tight  or  slack  side  of  the  transmission  may  be  estimated  by  sub- 


190  EFFICIENCY  OF  ROPE  DRIVES  [CHAP.  VIII 

stituting  the  proper  values  for  the  tension.  From  an  inspection 
of  (237),  it  is  evident  that  for  the  same  tension  Ta  in  the  rope  at 
A  there  are  two  different  values  of  h\'9  however,  in  rope-trans- 
mission problems  the  smaller  value  is  the  correct  one  to  use. 
The  statement  applies  equally  well  to  (238). 

It  is  important  to  note  that  the  above  discussion  applies  to  the 
rope  standing  still.  The  sag  of  a  rope  transmitting  power  may 
be  determined  approximately  by  means  of  (234)  by  substituting 
the  proper  value  of  the  tension  T. 

A  special  formula  may  be  deduced  for  the  case  in  which  the 
transmission  is  horizontal  having  sheaves  of  the  same  diameter. 

By  substituting  for  LI  =  ^  in  either  (237)  or  (238),  the  amount 
of  sag  h  is  given  by  the  following  expression  : 


In  general  the  bottom  rope  should  form  the  driving  side,  as 
with  this  arrangement  the  sag  of  the  slack  rope  on  top  increases 
the  arc  of  contact. 

148.  Efficiency  of  Manila  Rope  Drives.  —  The  efficiency  of 
manila  rope  transmission  is  generally  high  according  to  several 
series  of  experiments  performed  both  in  this  country  and  abroad. 
During  the  latter  part  of  1912,  the  Dodge  Mfg.  Co.  of  Misha- 
waka,  Ind.  conducted  a  series  of  experiments  to  obtain  some 
information  relating  to  the  efficiencies  of  four  general  plans  of 
manila  rope  driving.  The  four  plans  investigated  were  as 
follows  : 

1.  Open  drive  using  the  American  or  continuous  system,  as 
shown  in  Fig.  72. 

2.  Open  drive  using  the  English  or  multiple  system. 

3.  American  "up  and  over"  drive. 

4.  English  "up  and  over"  drive. 

In  the  tests  upon  these  various  plans  of  rope  driving,  from 
one-  to  eight-ropes,  operating  at  speeds  ranging  from  2,500  to 
5,500  feet  per  minute  were  used.  High-grade  manila  ropes  one 
inch  in  diameter,  treated  with  a  rope  dressing  so  as  to  make  them 
moisture-proof  and  to  preserve  the  surface,  were  used  throughout 
the  tests.  The  sheave  grooves  were  in  accordance  with  accepted 
Dodge  practice,  namely  a  60-degree  angle  for  the  American  sys- 
tem and  a  45-degree  angle  for  the  English  system,  All  idler 


ART.  148] 


EFFICIENCY  OF  ROPE  DRIVES 


191 


sheaves  used  in  the  various  arrangements  were  provided  with 
U-shaped  grooves. 


2500 


3000 


3500  4000  4500 

Rope  Speed  in  ft.  per  min. 
FIG.  76. 


5000 


5500 


Altogether  about  seven  hundred  tests  were  made,  the  general 
results  of  which  were  published  in  a  paper  presented  by  Mr.  E. 
H.  Ahara  before  the  American  Society  of  Mechanical  Engineers. 


100 


90 


80 


70 


60 


50 


17 


Eng.Open 


Amer.U.&O 


Eng  U.&O 


345 
tlumber     of     Ropes 
FIG.  77. 


An  analysis  of  the  results  published  seemed  to  indicate  that  the 
efficiency  for  low  rope  speeds  was  higher  than  that  obtained  at 
the  high  rope  speeds.  This  result  is  shown  clearly  in  Fig.  76, 


192  SELECTION  OF  ROPE  [CHAP.  VIII 

which  represents  the  results  obtained  from  the  tests  on  both 
systems  of  open  drive  operating  with  six  ropes  at  three-quarters 
load,  the  distance  between  the  centers  of  the  sheaves  being  fifty 
feet.  Furthermore,  the  tests  showed  that  the  efficiency  was  not 
affected  materially  by  varying  the  distances  between  the  driving 
and  driven  sheaves.  The  tests  also  showed  that  the  efficiency  at 
half  load  was  but  very  little  less  than  that  obtained  at  full  load. 
For  the  size  of  rope  used  in  the  experiments,  namely  one  inch, 
the  American  system  had  considerable  more  capacity  as  well  as 
a  higher  efficiency  than  the  English  system.  In  Fig.  77  is  repre- 
sented the  relation  existing  between  the  efficiency  and  the  num- 
ber of  ropes  used  for  the  four  plans  of  driving. 

149.  Selection  of  Rope. — Manila  ropes  for  transmission  pur- 
poses are  seldom  less  than  one  inch  in  diameter,  and  due  to  the 
resistance  offered  to  bending  over  the  sheaves,  ropes  exceeding 
one  and  three-quarter  inches  in  diameter  are  not  in  general  use. 
For  heavy  loads  such  as  are  met  with  in  rolling-mill  installations, 
ropes  two  inches  in  diameter  and  larger  are  used. 

In  order  to  arrive  at  the  proper  number  and  size  of  ropes 
required  to  transmit  a  given  horse  power,  the  size  of  both  the 
driving  and  driven  sheaves  should  be  decided,  as  the  smallest 
sheave  in  the  proposed  installation  will  determine  in  a  general 
way  the  largest  rope  that  may  be  used.  If  possible,  the  diame- 
ters of  these  sheaves  should  be  such  that  the  rope  will  operate  at 
somewhere  near  its  economical  speed,  which,  as  stated  in  Art. 
142,  has  been  found  in  practice  to  be  about  4,500  feet  per  minute. 
To  obtain  a  reasonable  length  of  service  from  a  given  rope,  its 
diameter  should  not  exceed  one-fortieth  of  the  diameter  of  the 
smallest  sheave.  According  to  the  American  Manufacturing  Co. 
of  Brooklyn,  N.  Y.,  it  is  considered  good  practice  to  use  a  small 
number  of  large  ropes  instead  of  a  large  number  of  small  ropes, 
notwithstanding  the  fact  that  the  first  cost  of  the  sheaves  for  the 
former  exceeds  that  required  for  the  smaller  ropes.  In  an  instal- 
lation using  a  small  number  of  large  ropes  the  number  of  splices 
is  smaller;  hence,  the  number  of  shutdowns  due  to  the  failure 
of  splices  is  decreased;  furthermore,  since  the  large  rope  has  a 
greater  wearing  surface,  its  life  is  increased. 

150.  Cotton  Rope  Transmission. — The  transmission  of  power 
by  means  of  cotton  rope  is  not  used  to  any  extent  in  this  country, 
but  in  England  it  is  used  extensively  in  all  kinds  of  installations. 


ART.  150]  COTTON  ROPE  193 

The  strength  of  good  cotton  rope  is  about  four-sevenths  of  that  of 
high-grade  manila  rope,  and  its  first  cost  is  about  50  per  cent, 
more.  Due  to  the  soft  fiber,  the  cotton  rope  is  more  flexible  than 
the  manila  rope,  and  for  that  reason  smaller  sheaves  may  be  used 
for  the  former.  According  to  well-established  English  practice, 
the  diameters  of  the  sheaves  are  made  equal  to  thirty  times  the 
diameter  of  the  rope.  The  cotton  rope,  as  generally  used,  is  com- 
posed of  three  strands,  and  being  somewhat  soft,  it  is  wedged  into 
the  grooves  of  the  sheave. 

According  to  some  of  the  American  rope  manufacturers,  a 
manila  rope  of  a  given  size  will  transmit  considerably  more  power 
than  the  same  size  of  cotton  rope.  In  view  of  this  statement 
it  is  interesting  to  compare  the  power  that  a  given  size  of  both 
manila  and  cotton  rope,  say  1J^  inches  in  diameter,  will  trans- 
mit at  a  speed  of  4,500  feet  per  minute.  According  to  a  well- 
known  American  manufacturer,  the  manila  rope  under  the 
above  conditions  will  transmit  29.1  horse  power.  According  to 
a  table  published  by  Edward  Kenyon  in  the  Transactions  of  the 
South  Wales  Institute  of  Engineers,  a  l^-inch  cotton  rope  will 
transmit  33.4  horse  power  at  the  same  speed.  This  result 
represents  an  increase  of  14.7  per  cent,  in  the  power  transmitted, 
and  also  indicates  that  higher  tensions  are  permissible  with 
cotton  rope.  As  stated  above,  cotton  rope  is  not  as  strong  as 
manila  rope;  hence,  these  higher  tensions  must  be  due  to  the 
structure  of  the  rope.  The  fibers  of  cotton  rope  being  soft  and 
more  flexible  do  not  cut  or  injure  each  other  when  the  rope  is 
subjected  to  bending  under  a  tension,  as  is  the  case  with  the 
manila  fiber;  the  grooves  of  the  cotton  rope  sheave  are  so  formed 
that  the  rope  is  wedged  into  the  groove  angle ;  hence,  the  effect  of 
centrifugal  force  is  not  so  marked  as  with  manila  rope  transmis- 
'S&b.  The  inference  is  clear  that  it  is  possible  to  employ  high 
speeds  with  cotton  rope;  and  such  is  the  case,  as  English  manu- 
facturers recommend  speeds  up  to  7,000  feet  per  minute. 

References 

The  Constructor,  by  F.  REULEAUX. 

Rope  Driving,  by  J.  J.  FLATHER. 

Machine  Design,  Construction  and  Drawing,  by  H.  J.  SPOONER. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Rope  Driving,  Trans.  A.  S.  M.  E.,  vol.  12,  p.  230. 

Working  Loads  for  Manila  Ropes,  Trans.  A.  S.  M.  E.,  vol.  23,  p.  125. 


194  REFERENCES  [CHAP.  VIII 

Efficiency  of  Rope  Drives,  Proc.  The  Eng'g  Soc.  of  W.  Pa.,  vol.  27,  No.  3, 
p.  73. 

Efficiency  of  Rope  Driving,  Trans.  A.  S.  M.  E.,  vol.  35,  p.  567. 

Transmission  of  Power  by  Manila  Ropes,  Power,  May  12,  1914  (vol.  39, 
p.  666). 

Transmitting  Power  by  Rope  Drives,  Power,  Dec.  8,  1914,  (vol.  40,  p.  808), 

The  Blue  Book  of  Rope  Transmission,  American  Mfg.  Co. 


CHAPTER  IX 
WIRE  ROPE  TRANSMISSION 

The  present-day  application  of  wire  rope  is  chiefly  to  hoisting, 
haulage,  and  transporting  service,  and  but  little  to  the  actual 
transmission  of  power.  In  this  chapter,  wire  rope  will  be  dis- 
cussed under  two  general  subheads  as  follows:  (a)  wire  rope 
hoisting,  and  (6)  wire  rope  transmission. 

WIRE  ROPE  HOISTING 

For  haulage  service,  the  six-strand  seven-wire  rope,  generally 
written  6  X  7,  is  used,  while  for  hoisting  a  6  X  19,  8  X  19,  or 
6  X  37  construction  is  employed.  The  rope  last  mentioned  is 
the  most  flexible  and  may  be  used  with  smaller  sheaves  than 
either  of  the  others,  but  the  wires  are  much  smaller;  hence  it 
should  not  be  subjected  to  excessive  external  wear.  The  6  X  19 
and  8  X  19  ropes  are  recommended  for  use  on  cranes,  elevators 
of  all  kinds,  coal  and  ore  hoists,  derricks,  conveyors,  dredges,  and 
steam  shovels.  The  6  X  37  rope,  which  is  extra  flexible,  is  used 
on  cranes,  special  hoists  for  ammunition,  counterweights  on 
various  machines,  and  on  dredges. 

A  hoisting  rope  under  load  is  subjected  to  the  following  prin- 
cipal stresses: 

(a)  Stresses  due  to  the  load  raised. 

(b)  Stresses  due  to  sudden  starting  and  stopping. 

(c)  Stresses  due  to  the  bending  of  the  rope  about  the  sheave. 

(d)  Stresses  due  to  slack. 

151.  Relation  between  Effort  and  Load. — In  hoisting  machinery 
calculations,  it  is  necessary  to  know  the  relation  existing  between 
the  effort  and  the  resistance  applied  to  the  ends  of  the  rope  run- 
ning over  a  sheave.  The  rigidity  of  the  rope  and  the  friction  of 
the  sheave  pin  increase  the  resistance  that  the  effort  applied  to 
the  running  off  side  must  overcome.  By  applying  the  same  line 

195 


196  BENDING  STRESSES  [CHAP.  IX 

of  reasoning  as  used  in  Art.  137,  we  obtain  a  relation  which  is 
similar  to  (207),  namely 

( 


The  efficiency  of  the  ordinary  guide  sheave,  obtained  by  apply- 
ing the  usual  definition  of  efficiency,  is  as  follows  : 

77  =  i  (241) 

152.  Stresses  Due  to  Starting  and  Stopping.  —  A  rope  whose 
speed  changes  frequently,  as  in  the  starting  and  stopping  of  a 
load,  is  subjected  to  a  stress  which  in  many  cases  should  not  be 
neglected.     This  stress  depends  upon  the  acceleration  given  to 
the  rope,  and  its  magnitude  is  determined  by  the  well-known 
relation,  force  is  equal  to  the  mass  raised  multiplied  by  the 
acceleration.     In  the  calculation  of  the  size  of  rope  for  mine 
hoisting  or  for  elevator  service,  the  stress  due  to  acceleration 
assumes  special  importance. 

153.  Stresses  Due  to  Bending.  —  The  stresses  due  to  the  bend- 
ing of  the  rope  about  sheaves  and  drums  are  of  considerable 
magnitude  and  should  always  be  considered  in  arriving  at  the 
size  of  a  rope  for  a  given  installation.     Several  formulas  for  calcu- 
lating these  stresses  have  been  proposed  by  various  investigators, 
but  they  are  all  more  or  less  complicated.     The  simplest  of  these 
is  the  following: 

Sb  =  E~>  (242) 

in  which  D  represents  the  pitch  diameter  of  the  sheave,  E  the 
modulus  of  elasticity  of  the  rope,  Sb  the  bending  stress  per  square 
inch  of  area  of  wires  in  the  rope,  and  8  the  diameter  of  the  wire  in 
the  rope.  This  formula  was  adopted  by  the  American  Steel  and 
Wire  Co.  To  determine  the  value  of  E  the  company  conducted 
a  series  of  experiments  on  some  six-strand  wire  rope  having  a 
hemp  center.  This  investigation  seemed  to  show  conclusively 
that,  the  modulus  of  elasticity  for  a  new  rope  does  not  exceed 
12,000,000.  Using  this  value  in  (242),  a  series  of  tables  was 
calculated  and  published  in  the  company's  Wire  Rope  Hand 
Book.  From  these  data  the  curves  shown  in  Figs.  78,  79  and 
80  were  plotted.  They  show  the  relation  between  the  bending 


ART.  153] 


BENDING  STRESSES 


197 


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BENDING  STRESSES 


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200 


STRESSES  DUE  TO  SLACK 


[CHAP.  IX 


stresses  and  various  diameters  of  sheaves  or  drums  for  the  more 
common  sizes  of  6  X  7,  6  X  19  and  6  X  37  wire  ropes.  How- 
ever, instead  of  using  the  ton  as  a  unit,  all  stresses  are  reduced 
to  pounds. 

154.  Stresses  Due  to  Slack. — In  any  kind  of  hoisting  operation 
it  is  important  that  the  rope  shall  have  no  slack  at  the  beginning 
of  hoisting,  else  the  load  will  be  suddenly  applied  and  the  stress 
in  the  rope  will  be  much  in  excess  of  that  due  to  the  load  raised. 
The  results  of  various  dynamometer  experiments  in  this  connection 
are  exhibited  in  Table  45. 

TABLE  45. — TENSIONS  DUE  TO  SLACK  AS  SHOWN  BY  DYNAMOMETER 


Weight  of  cage  and  load 

3,672 

6,384 

11,312 

11,310 

0 

4,032 

6,720 

11,542 

11,525 

5 

5,600 

11,200 

19,040 

19,025 

6 

8,960 

12,320 

23,520 

25,750- 

12 

12,520 

15,680 

28,000 

28,950 

The  theoretical  relation  between  the  tension  in  the  rope  and  the 
load  raised  may  be  deduced  as  follows: 

Let  W  =  load  to  be  raised. 

T  =  tension  in  the  rope  corresponding  to  the  maximum 

elongation. 

a  =  acceleration  of  the  rope  at  the  beginning  of  hoisting. 
b  =  elongation  of  the  rope  due  to  the  load  W. 
c  =  maximum  elongation  of  the  rope. 
e  =  amount  of  slack  in  the  rope. 

The  raising  of  the  rope  through  the  distance  e,  so  as  to  take  up 
the  slack,  may  be  considered  as  producing  the  same  effect  as 
dropping  the  load  W  through  the  distance  e,  assuming  the 
acceleration  in  both  cases  as  constant  and  equal  to  a.  Letting 
v  denote  the  velocity  at  the  instant  when  the  slack  e  is  taken  up, 
we  have  v2  =  2  ae.  From  this  it  follows  that  the  kinetic  energy 
of  the  load  W  at  the  instant  the  rope  is  taut,  is 

"_<&• 

Due  to  this  loading  the  rope  elongates  a  distance  c,  the  final 
tension  being  T.     Hence  W  in  moving  through  this  distance  c 


ART.  154]  STRESSES  DUE  TO  SLACK  201 

does  work  equal  to  We.  Immediately  preceding  the  elongation 
of  the  rope,  the  tension  therein  is  zero  and  at  the  end  of  the 
elongation  the  tension  has  a  magnitude  T;  therefore,  the  work 
of  the  variable  tension  during  the  period  of  rope  elongation  is 

Tc 

~2~.     To  do  this  internal  work,  the  load  has  given  up  its  kinetic 

energy  -    -  and  the  work  TFc;  hence 

Wc  (244) 


Assuming  that   Hooke's  Law  will  hold   approximately  in  the 
case  of  a  rope,  we  get 

T  =  ^  (245) 

Substituting  this   value   of    T  in  (244),  and  solving  for  c,  we 
finally  get 

c  =  b  ±  bJl  +  ^-  (246) 

The  conditions  of  the  problem  indicate  that  the  positive  sign 
is  the  proper  one;  hence,  substituting  the  value  of  c  in  (245), 


T  = 


If  the  slack  e  is  zero  (247)  shows  that  T  =  2  W  ;  that  is,  the 
tension  is  double  the  load,  which  fact  was  established  in  Art.  18. 
The  amount  of  slack  simply  has  the  effect  of  increasing  the  ratio 

T 
-ypj  which,  as  shown,  cannot  be  theoretically  less  than  two. 

It  is  not  to  be  expected  that  experiments  would  give  exactly  the 
theoretical  values,  on  account  of  the  fact  that  wire  rope  differs 
materially  from  a  rigid  rod,  and  a  certain  amount  of  stretch  not 
according  to  Hooke's  Law  will  come  into  play  before  the  actual 
elongation  of  the  material  begins.  This  fact  in  a  measure,  re- 
lieves the  "  suddenness,"  so  to  speak,  of  the  action,  and  we  would 
expect  the  tensions  measured  by  the  dynamometer  to  be  less  than 
those  given  by  (247).  To  get  the  experimental  values  by  means 
of  (247),  it  will  be  necessary  to  introduce  a  coefficient  K  in  the 
equation,  making  it 


=  W  \l  +  K    / 


(248) 


This  coefficient  must  of  course  be  determined  by  experiments, 
and  will  doubtless  vary  with  the  construction  of  the  rope  and  quite 


202  SELECTION  OF  ROPE  [CHAP.  IX 

likely  with  the  load  W  and  the  slack  e.  Unfortunately,  in  the 
experiments  quoted  in  Table  45  no  attempt  was  made  to  de- 
termine the  acceleration  of  hoisting,  and  as  a  consequence,  one 
essential  factor  is  lacking;  hence  it  is  impossible  to  arrive  at 
probable  values  of  the  coefficient  K  unless  an  assumption  regard- 
ing the  ratio  a  to  b  is  made. 

155.  Selection  of  Rope. — The  maximum  stress  coming  upon  a 
rope  is  the  summation  of  the  separate  stresses  that  may  be 
present  in  any  installation.     These  separate  stresses  have  been 
discussed  in  the  preceding  articles,  and  having  determined  their 
intensities,  the  magnitude  of  the  maximum  is  readily  obtained. 
The  next  step  is  to  determine  the  ultimate  strength  of  the  prob- 
able size  of  rope  to  be  used,  by  multiplying  the  maximum  stress 
by  a  factor  commonly  called  the  factor  of  safety.     This  factor 
varies,  with  the  class  of  service  for  which  the  rope  is  intended,  and 
the  following  values  may  serve  as  a  guide  in  the  solution  of  wire 
rope  problems : 

For  elevator  service  the  factor  of  safety  varies  from  8  to  12. 

For  hoisting  in  mines  the  factor  of  safety  varies  from  2%  to  5. 

For  motor  driven  cranes  the  factor  of  safety  varies  from  4  to  6. 

For  hand  power  cranes  the  factor  of  safety  varies  from  3  to  5. 

For  derrick  service  the  factor  of  safety  varies  from  3  to  5. 

Having  calculated  the  ultimate  strength,  select  the  size  of  rope 
that  is  strong  enough.  In  practically  all  hoisting  rope  calcula- 
tions, it  will  be  found  that  two  or  more  wire  ropes  of  different 
sizes  and  quality  will  satisfy  the  conditions  of  the  problem;  for 
example,  from  Table  46  it  is  evident  that  a  %-inch  crucible 
steel  rope  and  a  %-inch  plow  steel  rope  of  the  6  X  19  construc- 
tion have  the  same  ultimate  strength ;  hence,  either  of  these  ropes 
could  be  selected.  In  the  example  just  quoted,  the  %-inch  plow 
steel  rope  would  be  preferable  to  the  %-inch  crucible  steel  rope, 
since  the  smaller  sheave  called  for  by  the  former  size  would  effect 
a  saving  of  space  as  well  as  in  the  first  cost.  In  a  preceding 
paragraph,  the  uses  of  the  various  types  of  wire  rope  were  dis- 
cussed briefly.  In  Table  46  is  given  information  pertaining 
to  the  ultimate  strengths  and  weights  of  rope,  as  well  as  the 
minimum  diameter  of  sheaves  recommended  by  the  manufacturer. 

156.  Hoisting  Tackle. — The  analysis  of  blocks  and  tackles 
reefed  with  wire  rope  is  similar  to  that  given  in  Art.  138  for  manila 
rope,  and  the  formulas  deduced  there  also  apply  in  the  present 
case,  provided  a  proper  value  is  assigned  to  the  coefficient  C. 


ART.  156] 


WIRE  ROPE  TABLES 
TABLE  46. — STEEL  WIRE  ROPE 


203 


6X7  construction 

6  X  19  construction 

Diameter 
in  inches 

Weight 

Mini- 

Ultimate  strength 

Weight 

Mini- 

Ultimate  strength 

per 
foot 

mum 
sheave 
diam. 

Crucible 
steel 

Plow 
steel 

foot 

mum 
sheave 
diam. 

Crucible 
steel 

Plow 
steel 

K 

0.10 

12 

4,400 

5,300 

He 

0.15 

27 

7,000 

8,800 

0.15 

15 

6,200 

7,600 

*A 

0.22 

33 

9,200 

11,800 

0.22 

18 

9,600 

11,500 

K6 

0.30 

36 

11,000 

14,000 

0.30 

21 

13,000 

16,000 

K 

0.39 

42 

15,400 

20,000 

0.39 

24 

16,800 

20,000 

Ke 

0.50 

48 

20,000 

24,000 

0.50 

27 

20,000 

24,600 

K 

0.62 

54 

26,000 

32,000 

0.62 

30 

25,000 

31,000 

H 

0.89 

60 

37,200 

46,000 

0.89 

36 

35,000 

46,000 

K 

1.20 

72 

48,000 

62,000 

1.20 

42 

46,000 

58,000 

1 

1.58 

84 

62,000 

76,000 

1.58 

48 

60,000 

76,000 

IK 

2.00 

96 

74,000 

94,000" 

2.00 

54 

76,000 

94,000 

IK 

2.45 

108 

92,000 

120,000 

2.45 

60 

94,000 

116,000 

IK 

3.00 

120 

106,000 

144,000 

3.00 

66 

112,000 

144,000 

IK 

3.55 

132 

126,000 

164,000 

3.55 

72 

128,000 

164,000 

IH 

4.15 

78 

144,000 

188,000 

IK 

4.85 

84 

170,000 

224,000 

IK 

5.55 

96 

192,000 

254,000 

2 

6.30 

96 

212,000 

280,000 

2>£ 

8.00 

108 

266,000 

372,000 

2H 

9.85 

120 

340,000 

458,000 

2H 

11.95 

132 

422,000 

550,000 

Diameter 
in  inches 

8  X  19  construction 

6  X  37  construction 

Weight 
foot 

Mini- 
mum 
sheave 
diam. 

Ultimate  strength 

Weight 
per 
foot 

Mini- 
mum 
sheave 
diam. 

Ultimate  strength 

Crucible 
steel 

Plow 
steel 

Crucible 
steel 

Plow 
steel 

K 

0.20 

12 

9,320 

0.22 

12 

9,300 

10,600 

He 

0.27 

14 

12,600 

0.30 

14 

12,700 

15,000 

K 

0.35 

16 

16,000 

19,000 

0.39 

16 

16,500 

19,500 

He 

0.45 

18 

20,200 

24,000 

0.50 

18 

21,000 

25,000 

H 

0.56 

21 

24,800 

30,000 

0.62 

21 

25,200 

32,000 

K 

0.80 

22 

35,200 

44,000 

0.89 

22 

38,000 

46,000 

K 

1.08 

26 

46,000 

56,000 

1.20 

26 

50,000 

58,000 

1 

1.42 

30 

59,400 

72,000 

1.58 

30 

64,000 

74,000 

IK 

1.80 

34 

76,000 

92,000 

2.00 

34 

78,000 

92,000 

IK 

2.20 

38 

94,000 

112,000 

2.45 

38 

100,000 

116,000 

IK 

2.70 

42 

114,000 

136,000 

3.00 

42 

122,000 

142,000 

IK 

3.19 

45 

132,000 

160,000 

3.55 

45 

142,000 

168,000 

IK 

4.15 

158,000 

190,000 

l*/i 

4.85 

190,000 

226,000 

IK 

5.55 

212,000 

250,000 

2 

6.30 

234,000 

274,000 

2K 

8.00 

300,000 

368,000 

2H 

9.85 

374,000 

450,000 

2H 

11.95 

466,000 

556,000 

204 


HOISTING  SHEAVES 


[CHAP.  IX 


Experimental  data  on  wire-rope  hoisting  tackle. — Some  years 
ago  the  American  Hoist  and  Derrick  Co.  of  St.  Paul,  Minn., 
conducted  a  series  of  experiments  on  three  standard  sizes  of 
blocks  reefed  with  wire  rope.  The  results  of  these  tests  are  given 
in  Table  47.  By  using  the  relation  between  P  and  Q  in 
terms  of  C  for  the  various  combinations  listed,  it  is  possible  to 
calculate  the  value  of  C.  This  was  done  by  the  author  and  the 
values  are  tabulated  in  the  last  two  columns  of  Table  47. 


FIG.  81. 

An  inspection  of  the  values  of  C  given  in  this  table  shows  that  for 
a  given  size  of  rope  the  coefficient  C  may  safely  be  assumed  as 
constant. 

157.  Hoisting  Sheaves  and  Drums. — (a)  Sheaves. — The  sheaves 
used  for  hoisting  purposes  vary  considerably  in  their  design. 
For  crane  work  the  sheaves  are  usually  constructed  with  a  central 
web  in  place  of  arms  and  in  order  to  reduce  the  weight,  openings 
may  be  put  into  this  web.  Such  a  sheave  is  shown  in  Fig.  81 
and  in  Table  48  are  given  some  of  the  leading  dimensions 
pertaining  to  the  design  shown  in  Fig.  81.  As  a  rule,  sheaves  of 
this  class  are  bushed  with  bronze  or  some  form  of  patented  bush- 
ing, and  run  loose  on  the  pin.  For  very  heavy  crane  service,  the 
sheaves  are  frequently  made  of  steel  casting,  cast  iron  being 
used  for  the  medium  and  lighter  class  of  service. 


ART.  157]     EXPERIMENTAL  DATA  ON  HOISTING  TACKLE     205 
TABLE  47. — HOISTING  TACKLE  REEFED  WITH  WIRE  ROPE 


Size  of 
rope 

Block  and  tackle  data 

Ratio 
P/Q 

Value  of  0 

Sheave 
diam. 

Pin 

diam. 

No.  of 
sheaves 

No.  of 
lines 

Test 

Mean 

K 

9 

IK 

1 

2 

0.518 

1.075 

1.076 

2 

2 
3 

0.559 
0.358 

1.078 
1.076 

3 

3 
4 

0.385 
0.278 

1.076 
1.076 

4 

4 
5 

0.298 
0.230 

1.075 
1.076 

5 

5 

6 

0.247 
0.198 

1.076 
1.076 

6 

6 

0.213 

1.076 

H 

UK 

IK 

1 

2 

0.516 

1.068 

1.064 

2 

2 
3 

0.549 
0.355 

1.066 
1.066 

3 

3 
4 

0.376 
0.273 

1.063 
1.063 

4 

4 
5 

0.291 
0.225 

1.064 
1.063 

5 

5 
6 

0.240 
0.193 

1.064 
1.064 

6 

6 

0.206 

1.064 

% 

13% 

IK 

1 

2 

0.513 

1.053 

1.054 

2 

2 
3 

0.541 
0.351 

1.055 
1.054 

3 

3 
4 

0.369 
0.270 

1.053 
1.054 

4 

4 
5 

0.284 
0.221 

1.053 
1.054 

5 

5 
6 

0.233 
0.189 

1.054 
1.054 

6 

6 

0.199 

1.053 

For  heavy  high-speed  hoisting  as  found  in  mining  operations, 
the  arms  consist  of  steel  rods  cast  into  the  hub  and  rim,  as  shown 
in  Fig.  82.  Sheaves  of  this  class  are  not  bushed  as  in  crane  service, 
but  are  keyed  to  the  shaft. 

The  grooves  of  all  hoisting  sheaves  should  be  finished  smooth 


206  HOISTING  SHEAVES  [CHAP.  IX 

TABLE  48. — GENERAL  DIMENSIONS  OF  WIRE  ROPE  SHEAVES 


Dimensions  in  inches 

Size  of 

rope 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

u 

13-i 

M 

1 

KG 

K 

% 

% 

5/8 

iy8 

K 

IK 

« 

KG 

% 

1% 

H 

% 

2 

Ke 

IK 

KG 

KG 

% 

1% 

1M 

KG 

2% 

1 

K 

7/8 

2K 

% 

1% 

K 

% 

7/8 

IK 

2 

% 

2% 

IK 

^ 

1 

2K 

KG 

2 

KG 

% 

1 

m 

2K 

^6 

2% 

IK 

K 

IK 

2% 

KG 

2K 

% 

KG 

1H6 

2-2K 

2K 

H 

3K-4 

1M 

1 

so  as  to  protect  the  individual  wires  of  the  rope.  The  radius 
of  the  bottom  of  the  groove  should  be  made  slightly  larger  than 
the  radius  of  the  rope,  so  that  the  latter  will  not  be  wedged  into 


FIG.  82. 

the  groove.  It  is  important  that  the  alignment  of  sheaves  be  the 
best  possible,  otherwise  the  rope  will  slide  on  the  sides  of  the 
groove  and  cause  an  undue  amount  of  wear  on  both  the  rope  and 
sheave.  The  diameter  of  the  sheave  should  be  made  as  large 
as  possible  to  keep  down  the  bending  stresses.  In  Table  46 
are  given  the  minimum  sheave  diameters  recommended  by  the 
wire  rope  manufacturers.  It  is  customary  for  crane  builders  to 


ART.  157] 


HOISTING  DRUMS 


207 


use  much  smaller  sheaves.  By  using  sheaves  having  a  diameter 
of  from  eighteen  to  twenty  times  the  diameter  of  the  rope,  a 
considerable  saving  in  space  may  result  but  at  the  same  time 
the  life  of  the  rope  is  decreased  materially. 

(6)  Drums. — In  hoisting  machinery,  the  drums  are  usually 
grooved  to  receive  the  rope  and  their  lengths  should  be  sufficient 
to  hold  the  entire  length  of  rope  in  a  single  layer.  The  use  of  the 
plain  ungrooved  drum  should  be  avoided  unless  it  is  lagged.  If 
the  drum  is  grooved,  the  pitch  of  the  grooves  must  be  made 
slightly  larger  than  the  diameter  of  the  rope  so  that  the  successive 
coils  do  not  touch  when  the  rope  is  wound  onto  the  drum.  In 
Fig.  83  is  shown  the  form  of  groove  used  by  several  crane  builders, 


FIG.  83. 


and  in  Table  49  are  given  the  various  dimensions  required  to 
lay  out  these  grooves. 

TABLE  49. — DIMENSIONS  OF  GROOVES  FOR  WIRE  ROPE  DRUMS 


Size  of  wire  rope 


Dimen- 


sion 

% 

KG 

X 

KG 

% 

X 

7/s 

i 

1 

KG 

X 

KG 

X 

lHe 

13/16 

15/16 

IHe 

2 

%2 

X 

g/32 

KG 

% 

13/B2 

1X* 

% 

3 

%2 

K4 

X 

K4 

5/B2 

KG 

7/B2 

X 

The  diameters  of  the  drums  are  usually  made  larger  than  those 
of  the  sheaves  for  a  given  size  of  rope  in  order  to  keep  down  the 
length  to  a  reasonable  dimension.  In  general,  the  diameters  of 
crane  drums  vary  from  twenty  to  thirty  times  the  diameter  of 
the  rope.  The  speed  of  hoisting,  the  load  to  be  raised,  and  the 
life  of  the  rope  should  be  considered  in  arriving  at  the  proper 
diameter  of  the  drum.  In  order  to  relieve  the  rope  anchor  on 
the  drum,  always  add  about  two  extra  coils  to  the  calculated 


208 


CRANE  DRUMS 


[CHAP.  IX 


number,  so  that  the  extra  coils  of  rope  remain  unwound  on  the 
drum. 

158.  Design  of  Crane  Drums. — A  simple  design  of  a  plain 
hoisting  drum  running  loose  on  the  shaft  is  shown  in  Fig.  84. 


FIG.  84. 

The  shaft  a  is  driven  by  means  of  the  gear  b,  to  the  web  of  which 
are  bolted  the  double  conical  friction  blocks  c.  These  blocks  fit 
into  the  clutch  rim,  which  in  this  case  is  integral  with  the  drum 


FIG.  85. 

d.  To  rotate  the  drum  with  the  gear,  the  clutch  is  engaged  by 
sliding  the  drum  along  the  shaft  a  by  means  of  the  operating 
mechanism  shown  at  the  left.  This  design  is  used  on  light  hoist- 
ing engines  manufactured  by  the  Clyde  Iron  Works  of  Duluth, 
Minn. 


ART.  159]  CONICAL  DRUMS  209 

A  good  design  of  a  crane  drum  is  shown  in  Fig.  85.  In  this 
case  the  shaft  h  is  held  stationary,  the  drum  hubs  being  bushed 
with  bronze  as  shown.  The  driving  gear  m  is  keyed  rigidly  to 
the  drum  k,  which  in  this  case  is  scored  for  a  hoisting  chain  al- 
though the  same  design  of  drum  may  be  used  with  rope.  Fre- 
quently the  shaft,  instead  of  being  stationary,  is  cast  into  the 
drum  and  the  whole  combination  rotates  on  the  outer  bearings. 

The  correct  stress  analysis  for  a  hoisting  drum  is  a  complicated 
problem,  and  the  following  approximate  method  is  generally  used 
in  arriving  at,  or  for  checking,  the  thickness  of  the  metal  below 
the  bottom  of  the  groove: 

1.  Determine  the  bending  stresses  by  treating  the  drum  as  a 
hollow   cylindrical   beam   supported  at  the  ends.     Assume  the 
maximum  rope  loads  as  concentrated  at  or  near  the  middle, 
depending  upon  the  scoring  on  the  drum. 

2.  Determine  the  crushing  stress  due  to  the  tension  in  the  coils 
of  rope  about  the  drum.     The  rope  tension  varies  from  coil  to 
coil,   and  since  maximum  values  are  sought,  consider  only  the 
first  coil,  namely,  the  one  supporting  the  load. 

3.  Determine  the  shearing  stress  due  to  the  torsional  moment 
transmitted.     As  a  rule  this  stress  is  very  small  and  is  usually  not 
considered. 

4.  Combine   the   stresses   calculated   in    (1)    and    (2)    above. 
Drums  thus  designed  have  sufficient  strength,  and  in  general  the 
weight  is  not  excessive. 

159.  Conical  Drums. — In  mine  hoists,  it  is  a  usual  practice  to 
employ  drums  having  varying  radii  for  the  successive  coils  of  the 
rope.  The  object  of  such  an  arrangement  is  to  obviate  the  varia- 
tions in  the  load  on  the  drum  due  to  the  varying  length  of  rope. 
Theoretically,  the  net  moment  of  the  rope  pull  about  the  drum 
axis  should  be  a  constant  in  order  that  the  motors  or  engines 
coupled  to  the  drum  may  operate  economically.  This  condition 
would  require  a  drum  of  curved  cross-section,  a  form  that  would 
be  difficult  to  construct.  In  practice,  the  section  of  each  half  of 
the  drum  is  given  the  form  of  a  trapezoid,  and  for  that  reason  it 
is  possible  to  balance  the  moments  on  the  drum  at  but  two  points 
of  the  hoist,  namely  at  the  top  and  bottom. 

(a)  Relation  between  R2  and  Ri. — To  determine  the  relation  ex- 
isting between  the  large  and  small  diameters  R2  and  Ri  of  the 
drum,  so  as  to  fulfil  the  condition  just  mentioned,  we  may  pro- 
ceed as  follows,' 


210  CONICAL  DRUMS  [CHAP.  IX 

Let    C  =  weight  of  cage  and  empty  car. 
H  =  depth  of  mine  in  feet. 
Q  =  weight  of  ore  in  car. 
w  =  weight  of  the  hoisting  rope,  pounds  per  foot. 

Neglecting  the  inertia  forces,  the  moment  of  the  rope  tension 
at  the  beginning  of  the  hoisting  period  is 

Mi  =  (C  +  Q  +  wH)(l  +  M)#!  -  C(l  -  M)#2,      (249) 

in  which  the  symbol  n  represents  a  friction  coefficient  that  may 
be  assumed  as  equivalent  to  0.05  for  vertical  mine  shafts. 
The  moment  at  the  end  of  a  trip  is 

M*  =  (C  +  Q)(l  +  M)#2  -  (C  +  wH)(l  -  rfRi      (250) 


Equating  these  moments  and  solving  for  the  radius  of  the  drum 
at  the  large  end,  we  find 


Q  +  2C 


-| 


Evidently  the  greater  the  depth  of  the  mine  shaft,  the  greater 
is  wH  relative  to  Q  and  C,  and  the  greater  the  value  of  the 
factor  m. 

(6)  Length  of  the  conical  drum.  —  The  conical  drum  must  be 
provided  with  spiral  grooves  to  receive  the  rope,  and  the  number 
required  to  hoist  from  a  depth  H  is 


(252) 

Several  extra  turns  are  required,  so  that  at  the  beginning  of  hoist- 
ing the  rope  will  be  coiled  several  times  around  the  drum.  The 
same  number  should  be  added  at  the  end  of  hoisting.  This  num- 
ber of  extra  turns  is  fixed  by  state  mining  laws. 

If  L  denotes  the  length  of  the  drums  and  p  the  horizontal  pitch 
of  the  grooves,  then 

L  =  +  "'  »  (253) 


in  which  n'  represents  the  extra  number  of  coils  added. 

The  length  of  a  conical  drum  is  necessarily  great,  and  for  that 
reason  the  drum  must  be  located  at  a  considerable  distance  from 
the  mine  shaft  to  reduce  as  much  as  possible  the  angular  dis- 
placement of  the  rope  from  the  center  line  of  the  head  sheave. 


ART.  160]  FLAT  WIRE  ROPES  211 

This  displacement  is  called  the  fleet  angle  and  should  not  exceed 
one  and  one-half  degrees  on  each  side  of  the  center  line,  or  a  total 
displacement  of  three  degrees.  When  it  is  impossible  to  locate 
the  drum  far  enough  back  from  the  head  sheave  to  keep  the 
fleet  angle  within  these  limits,  it  is  necessary  to  guide  the  rope 
onto  the  head  sheaves  by  means  of  rollers  or  auxiliary  sheaves, 
(c)  Composite  drum. — For  deep  mines,  another  form  of  drum 
called  the  composite  drum  is  frequently  substituted  for  the  plain 
conical  type.  This  consists  of  a  cylindrical  center  portion  and 
conical  ends.  One  rope  is  wound  from  one  end  up  the  cone  and 
over  the  cylindrical  portion,  while  the  other  is  unwound  from  the 
cylindrical  part  and  down  the  other  cone.  This  form  of  drum 
has  the  advantage  of  decreased  diameter  and  shorter  length,  but 
possesses  the  disadvantage  of  not  entirely  balancing  the  effect  of 
the  rope. 

160.  Flat  Wire  Ropes. — In  the  preceding  articles,  the  round 
wire  rope  has  been  discussed  more  or  less  in  detail,  and  the  various 
points  brought  out  are  applicable  in  general  to  the  flat  rope. 
This  type  of  rope  consists  of  a  number  of  round  wire  ropes,  called 
flat  rope  strands,  placed  side  by  side.  Its  principal  uses  are  for 
mine  hoisting;  for  operating  emergency  gates  on  canals;  for  oper- 
ating the  spouts  on  coal  and  ore  docks;  and  in  elevator  service 
for  counterbalancing  the  hoisting  ropes.  The  individual  strands, 
composed  of  four  separate  strands  containing  seven  wires  each, 
are  of  alternate  right  and  left  lay  and  are  sewed  together  with 
soft  Swedish  iron  or  steel  wire.  The  sewing  wires,  being  much 
softer  than  the  wires  that  compose  the  strands,  serve  as  a  cushion 
for  the  strand  and  at  the  same  time  will  wear  out  much  faster. 
Flat  wire  rope  with  worn  out  sewing  may  be  resewed  with  new 
wire,  and  in  case  any  particular  strands  are  damaged,  they  may  be 
replaced  by  new  ones.  Flat  ropes  are  made  in  thicknesses  vary- 
ing from  3/£  inch  to  %  inch,  and  widths  ranging  from  1%  to  8 
inches.  The  material  used  in  the  construction  of  flat  ropes  may 
be  either  crucible  cast  steel  or  plow  steel,  the  former  being  more 
common.  The  following  are  some  of  the  advantages  flat  ropes 
possess  over  round  ropes. 

1.  In  hoisting  from  deep  mines  it  is  desirable  to  use  a  rope  that 
has  no  tendency  to  twist  and  untwist.     This  tendency  is  obvi- 
ated by  the  use  of  a  flat  rope. 

2.  The  reels  required  for  coiling  up  the  flat  rope  occupy  less 
space  and  are  much  lighter  and  cheaper  to  construct  than  large 


212  SINGLE  LOOP  SYSTEM  [CHAP.  IX 

cylindrical  and  conical  drums.  The  decrease  in  bulk  and  weight 
is  especially  important  when  the  mines  are  located  in  places 
accessible  only  by  pack  train. 

At  the  present  time  flat  ropes  are  used  but  little  for  mine  hoist- 
ing and  hence  the  field  of  application  of  such  ropes  is  more  or  less 
restricted. 

WIRE  ROPE  TRANSMISSION 

Wire  rope  as  a  medium  for  transmitting  power  is  used  where 
the  distances  are  too  great  for  manila  ropes.  The  recent  develop- 
ment of  electrical  transmission  is  gradually  crowding  out  the  wire 
rope,  though  for  distances  of  from  300  to  1,500  feet  it  is  consid- 
ered a  cheap  and  simple  method  of  transmitting  power.  Two 
systems  are  used,  namely,  the  continuous  or  endless  rope  used  in 
operating  cableways,  haulage  systems  and  tramways,  and  the 
single  loop,  the  latter  being  simply  a  modification  of  belt  driving. 

161.  Single  Loop  System. — To  transmit  power  by  means  of  a 
single  loop  with  a  minimum  amount  of  slippage,  a  certain  amount 
of  pressure  between  the  surfaces  in  contact  is  necessary.     This 
pressure  depends  upon  the  weight  and  the  tension  of  the  rope. 
Therefore,  for  short  spans  it  is  frequently  necessary  to  use  a  large 
rope  in  order  to  get  the  proper  weight,  although  the  tension  may 
be  increased  by  resplicing  or  by  the  introduction  of  a  tightener. 
The  last  two  methods  are  not  considered  good  practice  as  the 
rope  may  be  strained  too  much,  and  in  addition,  the  filling  in  the 
bottom  of  the  grooves  of  the  sheaves  wears  away  too  rapidly. 

Experience  has  shown  that  transmitting  power  by  means  of 
wire  rope  is  generally  not  satisfactory  when  the  span  is  less  than 
50  to  60  feet.  This  is  due  to  the  fact  that  the  weight  of  the  rope 
is  not  sufficient  to  give  the  requisite  friction  without  the  use  of 
tighteners.  When  the  distance  between  the  shaft  centers  ex- 
ceeds 400  feet,  successive  loops  are  used;  that  is,  the  driving 
sheave  of  the  second  loop  is  keyed  fast  to  the  shaft  of  the  driven 
sheave  of  the  first  loop,  or  double-groove  sheaves  may  be  used. 

162.  Wire  Transmission  Rope. — Wire  rope  used  for  transmit- 
ting power  consists  of  six  strands  laid  around  a  hemp  or  wire  core, 
each  strand  containing  seven  wires.     The  rope  with  a  hemp  core 
is  more  pliable  and  for  that  reason  is  generally  preferred  for  power 
transmission.     As  mentioned  in  a  preceding  paragraph,  large  ropes 
are  occasionally  required  to  get  a  satisfactory  drive,  and  in  such 


ART.  163] 


STRESSES  IN  WIRE  ROPE 


213 


installations  a  six-strand  nineteen- wire  rope  is  to  be  preferred. 
The  6X7  construction  of  rope,  having  much  larger  wires,  will 
stand  more  wear  than  the  6X19  construction,  but  requires  much 
larger  sheaves.  The  material  used  in  the  manufacture  of  wire 
transmission  rope  is  iron,  crucible  cast  steel,  and  plow  steel.  In 
Table  46  is  given  information  pertaining  to  two  kinds  of  high- 
grade  transmission  rope. 

163.  Transmission   Sheaves. — The  sheaves  for  transmission 
rope  are  quite  different  from  those  used  with  manila  rope,  as  will 
be  seen  by  consulting  Fig.  86.     The  grooves  are  made  V  shape 
with  a  space  below,  which  is  filled  with  leather, 

rubber,  or  hardwood  blocks.  One  prominent 
manufacturer  uses  alternate  layers  of  leather 
and  blocks  of  rubber  for  a  filling.  The  func- 
tion of  this  filling  is  to  increase  the  friction  be- 
tween the  rope  and  sheave  and  at  the  same 
time  reduce  the  wear  of  the  rope  to  a  mini- 
mum. The  filling  should  have  a  depression 
so  that  the  rope  will  run  central  and  not  come 
into  contact  with  the  iron  sides  of  the  grooves. 
The  speed  of  the  rim  of  the  sheave  should  not 
exceed  5,000  feet  per  minute. 

The  diameter  of  the  sheave  should  be  made 
as  large  as  practicable  consistent  with  the  per- 
missible rim  speed. 

Large  sheaves  decrease  the  bending  stresses 
and  at  the  same  time  increase  the  transmitting 
power  of  the  rope.     In  Table  46  are  given  the  minimum  diam- 
eter of  sheaves  that  should  be  used  with  the  various  sizes  of  6  X 
7  and  6  X  19  transmission  rope. 

164.  Stresses  in  Wire  Rope. — The  maximum  stress  in  a  wire 
rope  due  to  the  power  transmitted  should  always  be  less  than  the 
difference  between  the  maximum  allowable  stress  and  that  due 
to  the  bending  of  the  rope.     For  the  magnitude  of  the  bending 
stress  in  a  rope  running  over  a  sheave,  consult  Fig.  78.     As  the 
bending  stress  decreases,  the  load  stress  may  be  increased,  but 
the  sum  of  these  two  separate  stresses  should  never  exceed  from 
one-third  to  two-fifths  of  the  ultimate  strength  of  the  rope  given 
in  Table  46.     No  provision  is  made,  however,  for  the  weakening 
effect  of  a  splice  in  the  rope.     To  prevent  slippage  between  the 


FIG.  86. 


214 


SAG  OF  WIRE  ROPE 


[CHAP.  IX 


rope  and  the  sheave,  the  ratio  of  the  tight  to  the  loose  tension 
must  have  a  value  given  by  the  following  expression,  which  may 
be  derived  directly  from  (218)  by  making  the  angle  /3  equal  to 
90  degrees.  The  symbols  used  have  the  same  meaning  as  as- 
signed to  them  in  Art.  144. 


fTJ 


(254) 


For  the  coefficient  of  friction  ju,  Mr.  Hewitt  in  his  treatise  pub- 
lished by  the  Trenton  Iron  Co.,  recommends  the  values  given  in 
Table  50. 


TABLE  50. — COEFFICIENTS  OF  FRICTION  FOR  WIRE   ROPE 


( 

Condition  of  rop 

8 

Dry 

Wet 

Greasy 

Plain  groove 

0    170 

0  085 

0  070 

Wood-filled 

0  235 

0  170 

0  140 

Rubber-  and  leather-filled  

0.495 

0.400 

0.205 

To  determine  the  horse  power  capable  of  being  transmitted  by 
a  given  size  of  wire  rope  use  (221),  substituting  for//  in  that  equa- 
tion the  proper  value  from  Table  50.  As  in  the  case  of  manila 
ropes,  there  is  a  speed  that  makes  the  horse  power  transmitted 
a  maximum  and  beyond  which  the  horse  power  decreases.  To 
determine  the  speed  corresponding  to  the  maximum  horse  power, 
use  (222). 

165.  Sag  of  Wire  Rope. — The  question  of  sag  was  discussed  in 
Art.  147  in  connection  with  manila  ropes  and  the  various  formu- 
las deduced  also  apply  in  the  present  discussion.  It  is  desirable, 
whenever  possible,  to  make  the  lower  rope  of  a  transmission  do 
the  driving;  the  upper  or  slack  rope  sags,  thereby  increasing  the 
angle  of  contact  on  both  sheaves  and,  at  the  same  time,  the  trans- 
mitting capacity  of  the  installation.  According  to  the  Trenton 
Iron  Co.,  the  sag  of  the  tight  or  lower  rope  should  be  about  one- 
fiftieth  of  the  span,  and  that  of  the  slack  rope  about  double  this 
amount. 


ART.  165]  REFERENCES  215 

References 

The  Constructor,  by  F.  REULEAUX. 

Machine  Design,  Construction  and  Drawing,  by  H.  J.  SPOONER. 

Elements  of  Machine  Design,  by  W.  C.  UNWIN. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Die  Drahtseile,  by  J.  HRABAK. 

The  Application  of  Wire  Rope  to  Transportation,  Power  Transmission, 
etc.,  by  W.  HEWITT. 

Wire  Rope  Handbook,  by  American  Steel  and  Wire  Co. 
The  Transmission  of  Power  by  Wire  Rope,  Mine  and  Minerals,  April, 
1904. 


CHAPTER  X 
CHAINS  AND  SPROCKETS 

The  various  types  of  chains  found  in  engineering  practice  may, 
according  to  their  use,  be  grouped  into  the  following  classes : 
(a)  Chains  intended  primarily  for  hoisting  loads. 
(6)  Chains  used  for  conveying  as  well  as  elevating  loads, 
(c)  Chains  used  for  transmitting  power. 


HOISTING  CHAIN 


166.  Coil  Chain. — The  kind  of  chain  used  on  hoists,  cranes, 
and  dredges  is  shown  in  Fig.  87  (a)  and  is  known  as  coil  chain. 


The  links  are  made  either  of  an  elliptical  shape  or  with  the  sides 
parallel,  and  the  material  used  should  be  a  high-grade  refined 
wrought  iron  or  an  open-hearth  basic  steel.  A  chain  made  of  the 
latter  material  has  a  higher  tensile  strength  and  at  the  same  time 
stands  greater  abrasive  wear  than  one  made  of  wrought  iron.  In 
order  to  insure  flexibility  in  a  chain,  the  links  should  be  made 

216 


ART.  167] 


HOISTING  CHAINS 


217 


small.  A  small  link  has  the  added  advantage  that  the  bending 
action  at  the  middle  and  at  the  end  of  the  link  due  to  the  pull 
between  adjacent  links  is  decreased.  In  Table  51  are  given  the 
general  dimensions  of  the  link,  weight  per  foot,  and  the  approxi- 
mate breaking  strength  of  the  commercial  sizes  of  dredge  and 
crane  chains. 

TABLE  51. — HOISTING  CHAINS 


Size 

Pitch 

Weight 
per  foot 

Outside 
length, 
in. 

Outside 
width, 
in. 

Dredge  and 
shovel 

BBB  crane 

Approx.  breaking  load,  Ib. 

M 

2^2 

0.75 

IHe 

% 

5,000 

4,000 

5/l6 

% 

1.00 

1H 

IHe 

7,000 

6,000 

H 

% 

1.50 

1% 

IK 

10,000 

9,000 

KG 

Wa 

2.00 

2Ke 

•1H 

14,000 

13,000 

y* 

1% 

2.50 

2% 

1% 

18,000 

17,000 

%6 

1% 

3.25 

2% 

1% 

22,000 

20,000 

5/s 

1% 

4.00 

3 

2He 

27,000 

26,000 

lHe 

^He 

5.00 

3K 

2M 

32,500 

30,000 

H 

1% 

6.25 

3^ 

2K 

40,000 

36,000 

l*/16 

2He 

7.00 

3^ 

2% 

42,000 

40,000 

y* 

2%  6 

8.00 

4 

2% 

48,000 

44,000 

% 

2Ke 

9.00 

4% 

3K6 

54,000 

50,000 

i 

2K 

10.00 

4% 

3^ 

61,000 

57,000 

IKe 

2% 

12.00 

4K 

3He 

69,000 

65,000 

1M 

2M 

13.00 

5Ji 

3% 

78,000 

72,000 

1H6 

3Ke 

14.50 

5%  6 

3% 

88,000 

80,000 

1M 

3M 

16.00 

5M 

4^ 

95,000 

88,000 

IHe 

3% 

17.50 

6K 

4^ 

104,000 

96,000 

1% 

3%  6 

19.00 

6Ke 

4%6 

114,000 

104,000 

IKe 

3% 

21.16 

6^6 

4% 

122,000 

116,000 

1H 

3% 

23.00 

7 

5 

134,000 

124,000 

We 

4 

25.00 

7^ 

5^6 

142,000 

132,000 

1% 

4M 

28.00 

7M 

5K 

154,000 

144,000 

1H 

4M 

31.00 

8M 

5% 

166,000 

IK 

5M 

35.00 

9^ 

6% 

190,000 

2 

5M 

40.00 

10 

6% 

216,000 

2^ 

6M 

53.00 

UK 

7% 

273,000 

2M 

7 

65.00 

12M 

8% 

337,000 

2H 

7y* 

73.00 

13 

9^ 

387,000 

3 

m 

86.00 

14 

9% 

436,000 

167.  Stud -link  Chain. — A  type  of  chain  known  as  the  stud-link 
chain  is  shown  in  Fig.  87(6).     It  is  used  mainly  in  marine  work 


218 


CHAIN  DRUMS 


[CHAP.  X 


in  connection  with  anchors  and  moorings.  The  chief  advantage 
of  the  stud-link  chain  is  that  it  will  not  kink  nor  entangle  as 
readily  as  a  coil  chain.  Experiments  show  that  for  the  same  size 
of  link,  the  addition  of  the  stud  results  in  a  decrease  of  the  ulti- 
mate strength  of  the  chain.  An  analysis  of  the  stresses  in  chains 
shows,  however,  that  within  the  elastic  limit  the  stud-link  chain 
will  carry  a  much  greater  load  than  the  open-link  chain.  See 
Bulletin  No.  18  Univ.  of  Illinois  Experiment  Station,  G.  A.  Good- 
enough  and  L.  E.  Moore. 

168.  Chain  Drums  and  Anchors. — In  practically  all  cases 
where  short-link  chains  are  used  for  heavy  service,  as  on  cranes 
and  dredges,  drums  are  used  for  winding  up  the  chain.  Such 


(a) 


FIG.  88. 


drums  should  always  be  provided  with  machined  grooves.  Two 
forms  of  such  grooves  are  shown  in  Fig.  88,  and  the  dimensions 
given  in  Table  52  will  be  found  convenient  for  layout  purposes. 

TABLE  52. — DIMENSIONS  OF  GROOVES  FOR  CHAIN  DRUMS 


Type 

Di- 
men- 
sion 

Size  of  chair. 

H 

Me 

w 

91  6 

H 

Hie 

N 

>». 

H 

>M. 

1 

(a) 

a 
b 

K 

H? 

Me 

2He 

*% 

%, 

2^6 

2K 

3^ 

%6Z 

^. 

(6) 

a 
b 

\w  >- 
i-K  if>.\ 

05 

me 

K 

%2 

m 

Me 

^M, 

2Me 

H 

%, 

H. 

2^6 

'I?' 

The  diameter  of  the  drum  depends  upon  the  speed  of  hoisting, 
the  loads  to  be  raised,  and  the  life  of  the  chain.  For  close-link 
chain,  it  has  been  found  by  experience  that  the  drum  diameter 
should  not  be  made  less  than  twenty  times  the  thickness  of  the 


ART.  168] 


CHAIN  ANCHORS 


219 


chain  material,  and  it  is  better  to  make  it  about  thirty  times  the 
thickness  of  material.  If  the  drum  is  made  small  in  diameter 
relative  to  the  size  of  the  chain,  the  bending  action  on  the  link 


referred  to  in  Art.  166,  will  be  excessive,  thus  decreasing  the  life 
of  the  chain. 

The  drum  should  always  be  made  of  sufficient  length  so  that 
the  required  length  of  chain  may  be  wound  upon  it  in  a  single 
layer.  It  is  considered  good  design  to  have  one  or  two  coils  of 


220 


CHAIN  ANCHORS 


[CHAP.  X 


chain  remaining  on  the  drum  when  the  load  is  in  its  lowest  posi- 
tion, thus  reducing  the  stress  coming  upon  the  anchor.  The  cor- 
rect stress  analysis  for  a  hoisting  drum  is  rather  complex,  and  the 
approximate  method  outlined  in  Art.  158  for  a  drum  using  wire 
rope  is  applicable  to  chain  drums. 

Anchors. — The  method  of  anchoring  the  free  end  of  the 
chain  to  the  drum  should  be  given  attention.  In  Fig.  89  are 
shown  three  designs  taken  from  the  practice  of  several  crane 
builders.  The  method  shown  in  Fig.  89 (a)  is  faulty  for  the  fol- 
lowing reasons:  (1)  The  hole  for  the  tongue  of  the  anchor  is 


f   —i 


FIG.  90. 

drilled  in  the  chain  groove,  thus  increasing  the  bending  action  on 
the  tongue.  (2)  In  case  the  chain  should  ever  assume  the  posi- 
tion indicated  by  the  dotted  lines,  the  cap  screw  a  will  receive  the 
greatest  load  instead  of  the  tongue  of  the  anchor. 

The  design  shown  in  Fig.  89(6)  overcomes  the  first  objection' 
in  that  the  tongue  of  the  anchor  is  placed  in  a  hole  drilled  into 
the  solid  metal.  The  second  objection,  however,  also  applies  to 
this  design.  In  Fig.  89 (c)  is  shown  a  construction  that  is  cheap 
to  make  and  at  the  same  time  overcomes  both  objections.  In 
certain  designs  of  drums  it  may  not  always  be  as  convenient  to 
attach  an  anchor  of  this  type  as  one  of  the  first  two  types. 


ART.  169] 


CHAIN  SHEAVES 


221 


169.  Chain  Sheaves. — Sheaves  are  of  two  classes,  namely  those 
that  merely  guide  the  chain  as  in  changing  direction,  and  those 
that  are  fitted  with  pockets  to  receive  the  links  of  the  chain. 
The  latter  class  is  used  extensively  in  chain  hoists  in  place  of 
drums,  also  for  transmitting  power  under  certain  conditions. 

(a)  Plain  sheaves. — Designs  of  plain  sheaves,  referred  to  as  the 
first  class,  are  shown  in  Figs.  90  and  91.  The  proportions  of  the 
twp  types  of  sheaves  are  given  in  Table  53.  For  sheaves  of  large 
diameters",  arms  are  used,  while  with  small  sheaves  the  web  cen- 
ter has  given  better  satisfaction.  The  web  centers  may  be  plain, 


FIG.  91. 
TABLE  53. — DIMENSIONS  OF  PLAIN  CHAIN  SHEAVES 


Size  of 
chain 


Type—  Fig.  90 


Type—  Fig.  91 


H 


IHe 
IHe 


W 


H 


KG 
Me 


2% 


% 

IHe 
IHe 


He 


2 
2He 

2M 


222 


CHAIN  SHEAVES 


[CHAP.  X 


or  if  it  is  desirable  to  decrease  the  weight,  round  holes  may  be 
cut  out  as  shown  in  Figs.  90  and  91.  To  give  stiffness  to  the 
center  web,  the  side  ribs  as  shown  in  Fig.  90  are  added. 

(6)  Pocket  sheaves. — Sheaves  similar  to  the  one  shown  in  Fig. 
92  are  called  pocket  sheaves  and  are  used  principally  on  chain 
hoists  in  place  of  drums.  The  horizontal  links  fit  into  pockets 
cast  in  the  periphery  of  the  sheave,  while  the  vertical  links  fall 


FIG.  92. 

into  a  central  groove  as  shown.  In  order  to  design  a  sheave  of 
this  type,  the  various  calculations  involving  the  formulas  given 
below  must  be  carried  out  with  considerable  accuracy.  The 
dimensions  of  the  chain  and  the  number  of  pockets  or  teeth  T 
desired  enable  one  to  derive  a  formula  for  the  so-called  pitch 
diameter  D. 


ART.  169]  CHAIN  SHEAVES  223 

From  Table  51,  the  dimensions  a  and  b  of  Fig.  92  for  the  chosen 
size  of  chain  are  established,  and  from  the  number  of  pockets  T, 
we  find  for  the  angle  a, 

180° 
«  =  if-  (255) 

From  the  geometry  of  the  figure,  the  following  equations  are 
obtained: 

sin  (a  -  0)  =  |j  (256) 

sin  0  =  -^  (257) 

Eliminating  D  and  solving  for  0,  we  obtain 

tan  0  =       Sma  (258) 

COS  a  +  T 
o 

Since  the  dimensions  a  and  6  as  well  as  the  angle  a  are  known  in 
any  given  case,  (258)  is  used  to  determine  the  angle  0;  having 
determined  this  angle,  the  pitch  diameter  D  of  the  sheave  is  found 
by  means  of  (257). 

The  rim  of  the  pocket  sheave  may  be  proportioned  by  the  fol- 
lowing empirical  formulas,  in  which  d  and  w  denote  the  dimen- 
sions of  the  chain  links  as  given  in  Fig.  87.  Referring  to  Fig.  92: 

c  =  d  +  (>£"  to  KG") 

;.!?i<H»to"-'"         <259) 

The  thickness  of  the  web  should  not  be  made  less  than  the 
diameter  of  the  material  in  the  chain,  and  the  diameter  of  the 
hub  should  be  approximately  twice  the  diameter  of  the  pin  sup- 
porting the  sheave. 

Having  determined  the  pitch  diameter  and  the  general  pro- 
portions of  the  rim,  web,  and  hub  of  the  sheaves,  the  next  step 
is  to  make  a  full-size  drawing  with  the  pockets  from  %  to  Y±  inch 
longer  than  the  link.  The  layout  of  the  tooth  shown  in  Fig.  92 
represents  the  tooth  form  at  the  center  line  of  the  sheave  and 
not  at  the  side  of  the  central  groove  where  it  should  begin.  How- 
ever, since  the  pockets  are  to  be  made  somewhat  longer  than  the 
links,  the  tooth  may  be  given  this  form  at  the  side  of  the  central 


224 


RELATION  BETWEEN  P  AND  Q 


[CHAP.  X 


groove  and  it  will  be  found  that  sufficient  clearance  is  thus  pro- 
vided in  the  majority  of  cases.  The  face  of  the  tooth,  or  that 
part  lying  above  the  pitch  circle,  may  be  drawn  with  a  radius 
equal  to  three  or  four  times  the  diameter  of  the  chain  material. 

170.  Relation  between  P  and  Q. — When  a  chain  is  wound  on, 
or  unwound  from,  a  sheave,  the  relative  motion  between  the  links 
on  the  running-on  and  -off  sides  introduces  frictional  resistances. 
The  turning  of  one  link  in  another  is  similar  to  that  of  a  journal 
running  in  its  bearing,  and  the  relation  between  the  applied  effort 
P  and  the  resistance  Q  will  be  based  upon  the  theory  of  journal 
friction. 

In  Fig.  93  is  shown  diagrammatically  a  chain  raising  a  load  Q 
by  means  of  an  effort  P. 


FIG.  93. 

Let       D    =  pitch  diameter  of  the  sheave. 
Do  =  diameter  of  the  sheave  pin. 
d  =  size  of  the  chain. 
H  =  coefficient  of  journal  friction. 
Me  =  coefficient  of  chain  friction. 

On  the  load  side,  the  link  a  in  moving  from  the  position  E  to 
that  at  D  turns  through  the  angle  a  relative  to  the  link  b.  To 
overcome  the  frictional  resistance  between  these  links  requires  an 

amount  of  work  to  be  done  by  the  effort  P  equivalent  to  McQo«. 

£ 

At  the  same  time  that  the  link  a  is  moving  from  E  to  D,  a  link 
on  the  effort  side  is  running  off,  the  frictional  resistance  of  which 

requires  work  to  be  done  equivalent  to  fj.cP^a.     In  addition  to 


ART.  171]  CHAIN  BLOCK  225 

these  resistances,  the  friction  of  the  sheave  pin  must  be  overcome. 
For  the  loading  shown  in  Fig.  93,  the  pressure  on  the  pin  is 
(P  -f  Q) ;  therefore,  the  work  required  to  overcome  the  friction 
of  the  pin  for  an  angular  displacement  a  of  the  sheave  is  equiva- 
lent to  pa.  (P  +  Q)  ~ 

The  useful  work  done  is     ~    >  hence,  the  total  work  required 
by  the  effort  P  to  raise  the  load  Q  is 

from  which 


The  value  of  K  varies  from  1.04  to  1.10,  the  first  value  applying 
to  lubricated  chains  and  the  latter  to  chains  running  dry. 

Efficiency  of  chain  sheave. — By  applying  the  definition  of 
efficiency  to  the  case  discussed  above,  we  find  that  the  efficiency 
is 

-n  =  ^  (262) 

Introducing  the  values  of  K  given  above,  it  follows  that  the 
efficiency  of  a  chain  sheave  having  the  chain  lubricated  is  96  per 
cent.,  while  the  same  sheave  with  the  chain  running  dry  has  an 
efficiency  of  approximately  91  per  cent. 

171.  Analysis  of  a  Chain  Block. — With  the  aid  of  the  principle 
discussed  in  Art.  170,  it  is  a  simple  matter  to  analyze  blocks  reefed 
with  chains.  As  an  example,  it  is  required  to  determine  the  mag- 
nitude of  the  effort  P  that  is  required  to  raise  a  load  Q  by  means 
of  a  differential  chain  block,  similar  to  the  one  shown  diagram- 
matically  in  Fig.  94.  As  indicated  in  the  figure,  an  endless  chain 
is  reefed  around  the  compound  sheave  ab  and  the  lower  sheave  c. 
As  constructed,  this  block  is  always  made  self-locking,  except 
occasionally  when  the  chain  becomes  very  greasy,  the  load  will 
run  down.  Due  to  its  self-locking  property,  the  efficiency  is 
rather  low. 

(a)  Raising  the  load. — During  one  revolution  of  the  compound 
sheave,  the  part  d  of  the  chain  rises  a  distance  2  irR,  while  the 
part  e  descends  a  distance  2?rr;  hence  the  sheave  c  and  the  load 


226 


CHAIN  BLOCK 


[CHAP.  X 


Q  rise  a  distance  ir(R  —  r).     Without  friction,  the  work  of  the 
effort  is  2  TrPo-R ;  hence 

2  irPoR  = 
from  which 


p»  -  Q  a 

•*  o  —  2  {*• 


(263) 


in  which  n  denotes  the  ratio  ~-  Evidently  any  desired  reduc- 
tion may  be  obtained  by  varying 
the  difference  (R  —  r). 

Considering  the  lower  sheave 
c,  it  is  evident  from  (261)  that 
the  relation  between  the  tensions 
in  the  running-off  and  running-on 
chains  is  T\  =  KiT2,  where  K\ 
depends  upon  the  size  of  the  chain 
and  the  diameter  of  the  sheave 
c.  Furthermore,  Q  =  Tl  +  T2; 
hence,  the  following  expressions 
are  obtained : 


m      __ 


1  +  K 
Q  . 


(264) 


FIG.  94. 


Now  the  compound  sheave  ab 
is  held  in  equilibrium  by  the 
forces  P,  TI,  T2,  the  pin  reaction, 
and  the  friction  forces.  Taking 
account  of  friction,  we  then  have 

PR  +  TV  =  KtTiR  (265) 

where  K2  depends  upon  the  size  of  the  chain  and  the  diameter  of 
the  sheave  a. 

Combining  (264)  and  (265),  the  magnitude  of  the  effort  be- 


comes 


(266) 


Replacing   KI   and   Kz   by   an   average  value  denoted  by  K, 
(266)  becomes 


ART.  171]  CHAIN  BLOCK  227 

The  efficiency  for  the  differential  chain  hoist  is  given  by  the 
expression 


(b)  Lowering  the  load. — When  the  load  is  lowered,  the  frictional 
resistances  all  act  in  the  opposite  sense,  and  the  analysis  is  given 
by  the  following  equations: 


l  +  K 
KQ 


(269) 


Trf  =  K(P)R  + 

in  which  (P)  represents  the  pull  required  on  the  chain  so  as  to 
prevent  running  down  of  the  load.  Combining  the  three  equa- 
tions given  in  (269),  we  obtain  the  following  expressions  for  the 
effort  (P)  and  the  efficiency  (17)  for  reversed  motion  : 


(c)  Conditions  for  self-locking.  —  Whether  the  hoist  shown  in 
Fig.  94  is  self-locking  or  not  depends  upon  the  values  of  K  and 
n.  For  self-locking,  it  is  apparent  that  (P)  <0;  hence  the  crit- 
ical value  of  n  at  which  the  self-locking  property  commences  is 
given  by  the  equation, 


K     l+K     - 

from  which  it  follows  that 


1  -  nK*   <Q  (272) 

Therefore,  the  critical  value  of  the  ratio  -5  is 

K 

n  =  ~  (273) 

For  a  self-locking  hoist,     n  >  -^  (274) 

(d)  Experimental  data.  —  An  investigation  of  six  sizes  of  differ- 
ential chain  blocks  having  capacities  from  500  to  6,000  pounds, 
inclusive,  gave  actual  efficiencies  varying  from  28  per  cent,  for  the 
larger  capacities  to  38  per  cent,  for  the  smaller  sizes.  Further- 


228 


DETACHABLE  CHAIN 


[CHAP.  X 


more,  it  was  found  that  the  value  of  K  as  determined  from  equa- 
tion (267)  or  (268)  varied  from  1.054  to  1.09. 

CONVEYOR  CHAINS 

For  the  purpose  of  conveying  and  elevating  all  kinds  of  mate- 
rial, various  types  of  chains  are  used.  These  chains  may  be 
adapted  very  readily  to  a  wide  range  of  conditions  by  using  spe- 
cial attachments,  such  as  buckets  and  nights.  The  chains  used 
for  this  class  of  service  may  in  general  be  grouped  into  the  follow- 
ing two  classes:  (a)  detachable  or  hook-joint,  and  (6)  closed- 
joint. 

172.  Detachable  Chain. — The  detachable,  or  hook-joint  chain 
shown  in  Fig.  95  is  used  very  extensively,  and  under  favorable 
conditions  gives  good  service.  The  chain  shown  is  made  of 


FIG.  95. 

malleable  iron;  but  there  is  a  form  of  hook  chain  now  obtainable 
that  is  made  of  steel.  Since  the  joints  between  the  links  are  of 
the  hook  or  open  type,  this  kind  of  chain  is  not  well  adapted  to 
the  elevating  and  conveying  of  gritty  bulk  material;  however,  if 
the  joints  are  properly  protected,  slightly  gritty  material  may  be 
handled.  In  addition  to  this  class  of  service,  hook-joint  chains 
are  frequently  used  for  power-transmission  purposes  at  moderate 
speeds,  say  not  to  exceed  600  feet  per  minute  for  the  Ewart  chain 
shown  in  Fig.  95  and  a  considerably  higher  figure  for  the  lock 
steel  chain.  For  elevating  and  conveyor  service,  the  speeds 
seldom  exceed  200  feet  per  minute. 


ART.  173] 


TABLE  OF  EWART  CHAINS 


229 


173.  Strength  of  Detachable  Chain.— In  Table  54  is  given 
general  information  pertaining  to  the  standard  sizes  of  Ewart 
detachable  chain,  manufactured  by  the  Link  Belt  Co.  In  addi- 
tion to  the  sizes  listed,  a  large  number  of  special  sizes  are  made. 
In  order  that  a  chain  drive  may  be  durable,  a  proper  working  load 

TABLE  54. — EWART  DETACHABLE  CHAIN 


Chain  No. 

Approx.  links 
per  ft. 

Aver,  pitch 

Weight 
per  ft. 

Ultimate 
strength, 
Ib. 

25 

13.30 

0.902 

0.239 

700 

32 

10.40 

1.154 

0.333 

1,100 

33 

8.60 

1.394 

0.344 

1,190 

34 

8.60 

1.398 

0.387 

1,300 

35 

7.40 

1.630 

0.370 

1,200 

42 

8.80 

1.375 

0.570 

1,500 

45 

7.40 

1.630 

0.518 

1,600 

51 

10.40 

1.155 

0.707 

1,900 

52 

8.00 

1.506 

0.848 

2,300 

55 

7.40 

1.631 

0.740 

2,200 

57 

5.20 

2.308 

0.832 

2,800 

62 

7.30 

1.654 

1.022 

3,100 

66 

6.00 

2.013 

1.158 

2,600 

67 

5.20 

2.308 

1.196 

3,300 

75 

4.60 

2.609 

1.311 

4,000 

77 

5.20 

2.293 

1.456 

3,600 

78 

4.60 

2.609 

1.909 

4,900 

83 

3.00 

4.000 

1.944 

4,950 

85 

3.00 

4.000 

2.400 

7,600 

88 

4.60 

2.609 

2.438 

5,750 

93 

3.00 

4.033 

2.670 

7,500 

95 

3.00 

3.967 

3.000 

8,700 

103 

3.90 

3.075 

4.087 

9,600 

108 

2.55 

4.720 

3.570 

9,900 

110 

2.55 

4.720 

4.437 

12,700 

114 

3.70 

3.250 

5.180 

11,000 

122 

2.00 

6.050 

7.000 

15,000 

124 

3.00 

4.063 

6.666 

12,700 

146 

2.00 

6.150 

6.240       - 

14,400 

must  be  used.  This  depends  upon  the  speed  and  the  class  of 
service  for  which  the  chain  is  used.  After  a  considerable  number 
of  years  of  experimental  work,  the  Link  Belt  Co.  has  established  a 
series  of  factors  that  may  be  used  for  arriving  at  the  proper  work- 
ing stresses  at  various  speeds.  In  Fig.  96,  the  factors  just  re- 
ferred to  have  been  plotted  so  as  to  bring  them  into  more  con- 


230 


CLOSED-JOINT  CHAINS 


[CHAP.  X 


venient  form  for  general  use.  To  determine  the  working  stress 
for  any  particular  size  of  chain,  multiply  the  ultimate  strength  as 
given  in  Table  54  by  the  speed  coefficient  obtained  from  the  graph 
in  Fig.  96. 

174.  Closed-joint  Chains. — As  the  name  implies,  this  type  of 
chain  has  a  closed  joint;  because  of  this  fact  it  is  well  adapted  to 
the  elevating  and  conveying  of  gritty  and  bulk  material,  as  well 


•  0.1  1  -|  

r-0.ll  

/  N 

....                                     / 

\  

0.10    -1 

O.I? 

y 

S<-  

r 

0.09  -              —  / 

V 

0  13 

V-  

4-                                                          / 
c                                                     / 

•-          / 

r  

c 

£0.08    ^ 

V  

0.14'G 

<u                                —  71 
o                                     / 

V  

t 

0) 

"^0.07    -/ 
q>                        / 

\ 

o 
0.15-a 

(U 

Q.               -/ 

vn          T| 

s. 

\J) 

—  -7 

v  —  ~  — 

0.06  y( 

0.16 

0.05 

0.17 

700  600  500  400  300  200 

Speed    of     Chain  -    ft.    per  min. 

FIG.  96. 


100 


as  transmitting  power  at  moderate  speeds.  A  large  number  of 
different  types  of  closed-joint  chain  are  now  manufactured.  In 
Figs.  97,  98  and  99  are  shown  three  types,  the  first  two  being 
made  of  malleable  iron  and  the  third  of  steel.  The  closed-joint 
chains  are  made  in  the  same  sizes  as  the  detachable  chains;  hence 
the  sprockets  are  interchangeable.  In  the  better  grades  of  closed- 
joint  chains,  the  pins  and  bushings  used  are  frequently  made  of 
hardened  steel. 


ART.  174] 


CLOSED-JOINT  CHAINS 


231 


FIG.  97. 


© 


FIG.  98. 


FIG.  99. 


232 


STRENGTH  OF  CLOSED-JOINT  CHAINS        [CHAP.  X 


175.  Strength  of  Closed-joint  Chain. — The  information  given 
in  Table  55  pertains  to  the  chains  shown  in  Figs.  97  and  98.  The 
chain  shown  in  Fig.  97  is  manufactured  by  the  Link  Belt  Co.  and 
is  known  as  the  "400  Class  Closed-end  Pintle  Chain."  To 
determine  the  proper  working  load  for  this  chain,  the  ultimate 
strength  given  in  Table  55  must  be  multiplied  by  the  so-called 
speed  coefficient  mentioned  in  Art.  173,  values  of  which  may 
be  obtained  from  Fig.  96. 

TABLE  55. — CLOSED-JOINT  CONVEYOR  AND  POWER  CHAINS 


XjinK  licit  L^O  s.     4UU     t^iass 

Jettrey-lvley-Ubern  type 

d 

KM 

p4 

.D 

J3 

c 

*^ 

O 

,Q 

6 

1 

a 

I* 

a;    - 

d 

£ 

«3 

ee,  . 

<D     - 

to 

H 

s"" 

-SJ3 
c8  *> 

to 

X 

"S 

ft 

a£.S 

"S*» 

a 

°£ 

>-.' 

"M 

Ifj 

g  a 

a 

°ij 

^ 

M 

X   £> 

.gfl 

r* 

Q> 

g 

'3 

c3  £  C 

•^  « 

J3 

o*  *- 

s? 

.Bi 

o3   a> 

"-H  f^ 

o 

<%, 

Ji 

ss-'g 

Pta 

0 

<K 

<^ 

1 

p^ 

434 

8.6 

.398 

3,600 

25 

13.30 

0.903 

0.416 

1,000 

442 

8.8 

.375 

1.470 

5,900 

33 

8.60 

1.395 

0.525 

2,200 

445 

7.4 

.630 

1.428 

5,900 

34 

8.60 

1.395 

0.758 

700 

2,650 

445M 

7.4 

.630 

3,600 

42 

8.75 

1.370 

1.090 

3,000 

447 

7.4 

.630 

5,900 

45 

7.40 

1.623 

1.090 

3,300 

452 

8.0 
74 

.506 

con 

1    78"? 

7,000 

50 

12.00 

1.000 

0.548 

1,900 

462 

.  TC 

7.3 

.  OoU 

1.634 

1  .  t  oo 

2.314 

9,000 

52 

8.00 

1.517 

1.480 

4,750 

467 

5.2 

2.308 

1.336 

6,200 

55 

7.40 

1.630 

1.400 

4,925 

4,072 

7.3 

1.654 

2.445 

9,000 

57 

5.20 

2.307 

1.460 

5,800 

477 

5.2 

2.293 

1.960 

10,000 

62 

7.30 

1.647 

1.920 

5,850 

X477 

5.2 

2.308 

1.825 

7,300 

67 

5.20 

2.308 

1.680 

600 

6,000 

483 

3.0 

4.000 

15,000 

75 

4.60 

2.619 

1.970 

7,350 

488 

4.6 

2.609 

2.769 

12,000 

77 

5.20 

2.311 

1.740 

6,500 

488H 

4.6 

2.609 

13,000 

77^ 

5.20 

2.311 

2.320 

8,300 

41  nQ 

3Q 

Q    H7^ 

_      OQQ 

on  nnn 

t  luo 
4,124 

.  y 
3.0 

4.100 

33,000 

78 

4.60 

2.620 

2.170 

8,050 

78H 

4.60 

2.620 

2.880 

9,900 

83 

3.00 

3.970 

3.10Q 

11,425 

85 

3.00 

3.960 

3.950 

500 

13,500 

88 

4.60 

2.610 

2.630 

8,300 

103 

4.00 

3.058 

5.290 

13,530 

108 

2.55 

4.751 

5.180 

400 

14,800 

121 

2.06 

6.042 

3.600 

500 

16,600 

122 

2.00 

6.109 

8.510 

300 

24,980 

124 

3.00 

4.074 

11.770 

300 

40,000 

146 

2.00 

6.215 

8.120 

300 

23,500 

The  chain  shown  in  Fig.  98  is  manufactured  by  The  Jeffrey 
Mfg.  Co.  and  is  known  as  the  "Mey-Obern"  type.  The  proper 
working  stress  for  any  particular  speed  may  be  found  by  using 
the  speed  coefficients  given  in  Fig.  96. 

The  chain  shown  in  Fig.  99  differs  from  those  shown  in  Figs. 


ART.  175] 


TABLE  OF  UNION  STEEL  CHAINS 


233 


97  and  98  in  that  the  body  of  the  link  is  stamped  and  formed 
from  one  piece  of  steel,  the  sides  being  connected  across  the  top 
by  a  bridge  as  shown.  This  chain  is  manufactured  by  The  Union 
Chain  and  Mfg.  Co.  of  Seville,  Ohio,  and  is  made  in  two  types, 
namely,  the  bushing  type  and  the  roller  type.  The  information 
contained  in  Table  56  pertains  to  the  roller  type  shown  in  Fig. 
99,  the  upper  part  of  the  table  showing  the  commercial  sizes  used 
mainly  for  power  transmission,  while  the  lower  part  gives  the  sizes 

TABLE  56. — UNION  STEEL  CHAINS 


Driving  chains 

Chain 
No. 

Pitch 

Rollers 

Ultimate 
strength,  Ib. 

Length 

Diameter 

3R 

M 

H 
K 

5/8 

15/32 

3,500 

4R 

1 

1A 

5/8 

3A 

% 

5,000 

5R 
6R 

IK 

5/8 
% 
1 

K 

7,500 

W 

H 
1 

% 

10,500 

7R 

m 

i 

1 

14,000 

8R 

2 

1 

IK 

IK 

18,000 

Rollers 

bO 

Chain 
No. 

Approx. 
links 
per  ft. 

Average 
pitch 

Ultimate 
strength,  Ib. 

Length 

Diameter 

_g 

"« 

14 

8.0 

1.50 

K 

8,000 

g 

15R 

7.4 

1.62 

% 

1%6 

6,000 

1 

16R 

6.0 

2.00 

i 

IK 

12,000 

73 

17R 

5.2 

2.31 

i 

IK 

10,000 

08 

18R 

4.6 

2.61 

IK 

We 

12,000 

(3 

19 

3.9 

3.07 

1%6 

IK 

15,000 

DQ 

21 

3.4 

3.51 

IK 

1% 

22,000 

22 

3.0 

4.00 

IK 

1% 

30,000 

30 

2.0 

6.00 

IK 

1% 

40,000 

234 


CHAIN  SPROCKETS 


[CHAP.  X 


that  have  been  designed  to  run  on  standard  sprockets  used  for 
detachable  chain.  The  latter  type  of  chain  is  used  for  either 
power  or  conveyor  service.  To  arrive  at  the  working  stress  for 
a  given  speed,  multiply  the  ultimate  strength  given  in  Table  56 
by  the  speed  coefficient  taken  from  the  graph  in  Fig.  96. 

176.  Sprockets  for  Detachable  Chains. — Cast  sprockets  are 
generally  inaccurate  due  to  shrinkage  and  rapping  of  the  pattern; 
hence  in  order  to  get  satisfactory  service  they  should  be  made  a 


FIG.   100. 

trifle  large  and  then  ground  to  fit  the  chain.  The  sprockets  made 
from  ordinary  cast  iron  give  good  service,  especially  if  both  the 
chain  and  the  face  of  the  sprockets  are  lubricated  with  a  heavy 
oil  or  a  thin  grease.  For  severe  service  manufacturers  furnish 
sprockets  having  chilled  rims  and  teeth,  while  the  hubs  are  soft 
for  machining  purposes. 

Armor-clad  sprocket. — Another  form  of  sprocket  that  is  in- 
tended to  give  great  durability  is  shown  in  Fig.  100.  It  consists 
of  a  cast-iron  central  body  in  the  periphery  of  which  are  milled 
slots.  Into  these  slots  are  fitted  the  teeth  a,  which  are  formed 


ART.  177] 


CHAIN  SPROCKETS 


235 


from  special  steel  strips.  To  fasten  the  teeth  rigidly  in  the  body, 
the  ends  are  expanded  by  means  of  a  steel  pin  6,  and  lateral  dis- 
placement is  prevented  by  washers  and  riveting.  The  teeth  are 
heat  treated  and  may  be  removed  very  readily.  This  design  of 
sprocket  is  used  by  The  Union  Chain  and  Mfg.  Co. 

It  is  claimed  that  these  sprockets,  and  also  sprockets  having 
chilled  rims  and  teeth,  are  more  economical  since  they  last  con- 
siderably longer,  although  they  cost  approximately  50  per  cent, 
more  than  the  gray  iron  sprockets. 

177.  Relation  between  Driving  and  Driven  Sprockets. — Theo- 
retically the  pitch  of  the  sprocket  teeth  and  that  of  the  chain 
should  be  exactly  the  same;  but  as  chains  may  vary  a  trifle  from 


FIG.   101. 

the  exact  pitch,  and  as  the  wear  of  the  joints  tends  to  lengthen 
the  pitch,  some  provision  must  be  made  to  take  care  of  this  elon- 
gation or  the  chain  will  ride  on  the  teeth  of  the  sprocket.  To 
overcome  this  riding  action,  the  teeth  of  sprockets  are  given  back 
clearance;  that  is,  their  thickness  on  the  pitch  circle  is  made  less 
than  the  dimension  s  shown  in  Figs.  95,  97,  98,  and  99.  Further- 
more, the  pitch  of  the  teeth  is  increased  or  decreased,  depending 
upon  whether  the  sprocket  is  the  driving  or  driven  member  of 
the  transmission. 

In  Fig.  101  are  shown  two  sprockets  transmitting  power,  a 
being  the  driver  and  b  the  driven.     This  figure  shows  the  correct 


236  SPROCKET  TOOTH  FORM  [CHAP.  X 

chain  action,  and  it  should  be  noted  that  on  each  sprocket  the 
entire  load  transmitted  by  the  chain  comes  upon  one  tooth, 
namely  upon  the  one  at  the  point  where  the  links  run  off  the 
sprocket.  Referring  to  Fig.  101,  it  is  evident  that  the  loaded 
tooth  on  the  driving  sprocket  in  pushing  the  chain  forward  per- 
mits the  disengaging  link  to  roll  out  to  the  tip  of  this  tooth  and 
at  the  same  time  the  chain  creeps  backward  a  distance  equal  to 
the  increment  x.  By  the  time  the  driving  tooth  is  completely 
disengaged,  the  following  tooth  of  the  wheel  is  seated  firmly 
against  the  following  link ;  hence  it  follows  that  the  chordal  pitch 
pi  of  the  sprocket  is  greater  than  the  pitch  of  the  chain.  A  simi- 
lar analysis  of  the  action  of  the  chain  on  the  driven  sprocket  shows 
that  the  disengaging  link  in  rolling  out  on  the  loaded  tooth  creeps 
ahead  a  distance  equal  to  the  increment  y,  thus  bringing  the  follow- 
ing link  and  tooth  into  intimate  contact.  It  is  evident,  therefore, 
that  the  chordal  pitch  p2  of  the  sprocket  b  should  be  less  than  the 
pitch  of  the  chain.  The  condition  may  also  be  met  by  making 
the  chordal  pitch  p2  of  anew  sprocket  equal  to  the  chain  pitch,  and 
as  soon  as  the  wear  appears  the  links  creep  away  from  the  teeth 
producing  the  action  just  discussed. 

Sprockets  laid  out  as  shown  in  Fig.  101  are  likely  to  show  ex- 
cessive wear  since  one  tooth  must  carry  the  entire  load  trans- 
mitted by  the  chain.  According  to  information  furnished  by 
The  Jeffrey  Mfg.  Co.,  the  amount  that  the  driving  sprocket  is 
made  larger  than  the  theoretical  size  depends  upon  the  pitch  of 
the  chain,  the  size  of  roller  or  hook  of  the  link,  the  strength  of  the 
chain,  and  the  number  of  the  teeth  in  the  sprocket. 

178.  Tooth  Form. — From  the  discussion  given  in  Art.  177,  it  is 
evident  that  the  teeth  of  sprockets  must  be  given  considerable 

clearance  so  as  to  permit  the  chain 

TABLE  57.— SPROCKET    TEETH     to  elongate  due  to  the  load  as  well 
'^ as  the  wear  on  the  pins  and  not  per- 


No.  of  teeth 


8  to  12 

13  to  20 
21  to  35 
36  to  60 


Factor  mit  it  to  ride  on  the  flanks  of  the 

teeth.     If   the    chain   transmission 
0 . 75  to  0 . 80       is    designed   properly,    each   tooth 
comes  into   action   only  once   per 


0.65 
0.55  to  0.60 


revolution  of  the  sprocket;  hence, 
in  sprockets  having  large  numbers 
of  teeth,  the  wear  on  the  tooth 

flanks  is  distributed  over  more  teeth,  and  for  that  reason  the 
thickness  of  the  tooth  at  the  pitch  line  may  be  made  less  than 


ART.  178] 


SPROCKET  TOOTH  FORM 


237 


in  smaller  sprockets.  The  data  included  in  Table  57  will 
serve  as  a  guide  in  laying  out  the  teeth  of  sprockets.  To  ob- 
tain t,  the  thickness  of  the  tooth  at  the  pitch  line,  for  any  given 
size  of  chain  multiply  the  length  of  the  available  tooth  space  in 
the  link  by  the  factors  given  in  the  table.  These  factors  represent 
the  practice  of  the  Link  Belt  Co.  and  are  based  upon  experience 
with  chains  in  service.  By  the  available  tooth  space  in  the 
link  is  meant  the  dimension  s  in  Figs.  95,  97,  and  98. 

Having  decided  upon  the  size  of  chain  and  the  number  of  teeth 
in  the  sprocket  for  the  particular  case  under  consideration,  deter- 


FIG.  102. 

mine  the  pitch  diameter  of  the  sprocket  by  the  following  ex- 
pression : 

D  =  -f-9  (275) 

Sin  a 

in  which  D  denotes  the  pitch  diameter,  p  the  pitch  of  the  chain, 
and  a  equals  180  degrees  divided  by  the  number  of  teeth  in  the 
sprocket. 

Having  calculated  the  pitch  diameter,  the  sprocket  teeth  may 
be  laid  out  as  shown  in  Fig.  102.  The  root  circle  diameter,  as 
shown  in  the  figure,  is  fixed  by  the  dimensions  of  the  link.  An 
examination  of  a  considerable  number  of  sprockets  made  by  lead- 
ing manufacturers  seems  to  indicate  that  the  outline  of  the  tooth 
may  be  made  a  straight  line  between  the  root  circle  and  the 
rounded  corner  at  the  top.  The  radius  r  of  this  corner  varies 
from  KG  mcft  f°r  small  chains  to  about  %  inch  for  the  larger 


238  PROPORTIONS  OF  SPROCKETS  [CHAP.  X 

chains.  The  inclination  of  this  line  must  provide  sufficient 
clearance  to  prevent  interference  between  the  tooth  and  the  link 
when  the  latter  is  entering  or  leaving  the  sprocket.  The  flank 
of  the  tooth  is  joined  at  the  root  circle  by  a  fillet  having  a  radius 
less  than  that  of  the  hook  of  the  link. 

179.  Rim,  Tooth,  and  Arm  Proportions. — (a)  Rim  and  tooth. — 
The  rim  of  the  sprocket  may  be  proportioned  in  a  general  way  by 
the  following  empirical  formulas  taken  from  Halsey's  Handbook 
for  Machine  Designers  and  Draftsmen.  In  these  formulas  the 
dimensions  denoted  by  p  and  w  are  obtained  from  the  size  of  the 
chain  under  consideration. 


c  =  0.5  w 

I^{Q  w  for  small  chains 
w  —  J'g"  for  large  chains 


a  = 


e  =  %  w 


g  =  0.7  w 

k  =  1.25    (p  -  s) 


(276) 


(6)  Arm  proportions. — Sprockets  are  made  with  a  web  center 
or  with  arms.  For  very  small  pitch  diameters,  solid  web  centers 
having  a  thickness  determined  by  the  dimension  a  in  Fig.  102 
should  be  used.  For  larger  diameters  up  to,  say,  12  or  15  inches, 
web  centers  with  holes  may  be  used;  but  in  these  cases  the  web 
thickness  should  be  made  equal  to  approximately  six-tenths 
of  the  dimension  a  as  determined  by  means  of  (276).  For  diam- 
eters exceeding  12  or  15  inches,  the  sprockets  should  be  con- 
structed with  arms,  the  dimensions  of  which  may  be  obtained 
by  the  following  analysis: 

Let  W  =  breaking  load  of  the  chain. 

S  =  permissible  working  stress  for  the  material. 
b  =  thickness  of  the  arm  at  the  center  of  the 

shaft. 

h  =  depth  of  the  arm  at  the  center  of  the  shaft. 
n  =  number  of  arms,  4  to  6. 

To  be  on  the  safe  side,  the  arm  of  the  sprocket  is  designed  for  a 
load  exceeding  that  coming  upon  the  chain.  This  condition  is 
met  by  assuming  that  one-fifth  of  the  breaking  load  of  the  chain 
comes  upon  the  arms.  Equating  the  bending  moment  per  arm 
to  its  resisting  moment,  considering  the  arm  to  be  extended  to  the 


ART.  180] 


BLOCK  CHAINS 


239 


center  of  the  shaft,  we  have,  assuming  the  arm  to  have  an  ellipti- 
cal cross-section, 

WD 


10  n 


32 


from  which 


bh2  =  — ^-(approximately) 


(277) 


The  arms  of  sprockets  are  generally  made  with  a  cross-section 
approximating  an  ellipse  having  a  ratio  between  the  major  and 
minor  axis  of  about  2.5  to  1  at  the  center  of  the  sprocket.  At  the 
rim,  the  major  and  minor  axes  are  made  0.8  and  0.3,  respectively, 
of  the  major  axis  at  the  center. 

An  investigation  of  actual  sprockets  based  upon  the  above 
assumptions  showed  that  S  varied  from  2,500  to  3,300  pounds 
per  square  inch,  in  round  numbers.  As  an  average  value  use 
3,000.  Letting  6  =  0.4  h,  (277)  becomes 


h  = 


(278) 


POWER  CHAINS 


The  types  of  chains  discussed  in  the  preceding  articles  of  this 
chapter  are  not  well  adapted  to  any  service  requiring  speeds 


FIG.  103. 

above  600  feet  per  minute,  and  for  that  reason  they  are  not  suit- 
able for  the  transmission  of  power  where  the  speed  exceeds  this 
limit.  For  this  class  of  service  special  forms  of  chains,  all  parts 
of  which  are  machined  fairly  accurately,  have  been  devised. 
These  may  be  classified  as  follows:  (a)  block  chains;  (6)  roller 
chains;  (c)  silent  chains. 

180.  Block  Chains. — As  the  name  implies,  the  block  chain, 
shown  in  Fig.  103,  consists  of  solid  steel  blocks  shaped  like  the 


240 


TABLE  OF  DIAMOND  BLOCK  CHAINS         [CHAP.  X 


letter  B  or  the  figure  8,  to  which  the  side  links  are  fastened  by 
hardened  steel  rivets.  Block  chains  have  proven  very  satis- 
factory for  light  power  transmission  where  the  speeds  do  not  ex- 
ceed 800  to  900  feet  per  minute.  Table  58  gives  the  commercial 
sizes  of  the  block  chains  manufactured  by  the  Diamond  Chain 
and  Mfg.  Co. 

TABLE  58. — DIAMOND  BLOCK  CHAINS 


Chain  No. 

Pitch 

Dimensions 

Width 
of 
block 

Diam. 
of 
rivet 

Weight 
per 
foot 

Ultimate 
strength 

a 

b 

h 

y* 

% 

0.33 

1,500 

102 

1 

0.400 

0.600 

0.325 

%6 

% 

KG 

0.38 
0.42 

1,600 
1,800 

y* 

0.50 

2,000 

% 

*K4 

0.33 

2,200 

103 

1 

0.400 

0.600 

0.325 

KG 

H 

] 

KG 

0.38 
0.42 

2,300 
2,400 

l/2 

0.50 

2,500 

105 

IK 

0.564 

0.936 

0.532 

H 

5/s 

0.265 

0.89 
1.03 

5,000 

181.  Sprockets  for  Block  Chains. — In  Fig.  104  is  shown  a 
design  of  a  block  chain  sprocket,  the  rim  part  being  made  of  steel 
plate  bolted  on  to  a  cast-iron  hub.  Instead  of  using  the  built- 
up  construction,  the  sprocket  may  be  made  completely  of  cast 
iron  with  a  central  web,  or  with  arms,  if  the  sprocket  is  large  in 
diameter.  Denoting  the  pitch  diameter  of  the  sprocket  by  the 
symbol  D,  the  number  of  teeth  in  the  sprocket  by  T,  and  the 
pitch  of  the  chain  by  p,  then  the  magnitude  of  the  angle  a  shown 
in  Fig.  104  is  given  by  the  following  expression: 


a  = 


180C 


(279) 


From  the  geometry  of  the  figure,  it  follows  that 
sin  (a  -  j8)  =  ^ 


(280) 


and 


sin  p  == 


(281) 


ART.  181] 


BLOCK  CHAIN  SPROCKETS 


241 


Deriving  an  expression  for  $  by  eliminating  D,  we  have 

sin  a. 


tan  |8 


.    a 

cos  a  +  T 
o 


(282) 


To  obtain  the  pitch  diameter  of  the  sprocket  for  any  desired 
number  of  teeth  and  given  size  of  chain,  determine  the  angle  a 
and  substitute  this  angle  in  (282)  in  order  to  establish  the  angle 
|8.  Knowing  0,  the  pitch  diameter  D  may  be  found  by  means  of 
(281).  To  get  satisfactory  service  from  sprockets,  the  minimum 


FIG.  104. 

number  of  teeth  should  be  limited  to  15,  unless  the  rotative 
speed  of  the  sprocket  is  low.  The  teeth  of  small  sprockets  have 
a  tendency  to  wear  hook-shaped,  thus  causing  noise  and  at  the 
same  time  decreasing  the  life  of  the  installation. 

The  height  of  the  tooth  is  usually  made  slightly  greater  than 
the  dimension  h  in  Table  58.  It  should  be  noted  that  the  space 
between  the  teeth  is  made  somewhat  longer  than  the  overall 
length  of  the  block,  in  order  to  provide  for  the  stretching  of  the 
chain  due  to  wear  on  the  rivets. 


242  SELECTION  OF  BLOCK  CHAINS  [CHAP.  X 

182.  Selection  of  Block  Chains. — A  careful  study  of  the  opera- 
tion of  chains  of  the  block  and  roller  type  conducted  by  the 
Diamond  Chain  and  Mfg.  Co.  indicates  that  the  noisy  operation 
and  the  rapid  wear  of  a  chain  are  due  chiefly  to  the  impact  be- 
tween the  sprocket  and  the  rollers  or  blocks  as  the  latter  seat 
themselves.  The  effect  of  impact  is  more  marked  when  a  long 
pitch  chain  runs  over  a  sprocket  having  a  high  rotative  speed. 
As  a  result  of  this  study,  the  following  empirical  formulas  and 
rules  have  been  proposed  by  the  Diamond  Chain  and  Mfg.  Co. : 


900 


max.  p 
max.  N  of  small  sprocket  =  — 

vV 


(283) 


(a)  In  an  installation  in  which  the  load  on  the  chain  is  fairly 
uniform,  the  permissible  chain  pull  should  not  exceed  one-tenth 
of  the  ultimate  strength  of  the  chain  as  given  in  Table  58. 

(6)  As  a  further  check  on  the  chain  load,  the  equivalent  pres- 
sure per  square  inch  of  projected  rivet  area  should  not  exceed 
1,000  pounds  for  general  service.  When  slow  chain  speeds  pre- 
vail, this  pressure  may  run  as  high  as  3,000  pounds,  although  the 
latter  value  should  be  considered  the  upper  limit. 

(c)  When  the  chain  is  subjected  to  sudden  fluctuations  of  load, 
the  permissible  chain  pull  may  only  be  J-^o  or  /4o  of  the  ultimate 
strength. 

(d)  In  selecting  a  block  or  roller  chain  for  a  given  duty  it  is  well 
to  give  preference  to  a  light  chain  rather  than  a  heavy  one,  pro- 
vided the  former  has  sufficient  rivet  area  as  well  as  strength  to 
transmit  the  power.     As  stated  above,  long  life  and  quiet  run- 
ning are  secured  more  easily  by  selecting  a  short  pitch  chain. 
As  a  rule,  a  narrow  chain  is  more  satisfactory  than  a  wide  one 
except  in  places  where  the  sprockets  are  not  always  in  proper 
alignment;  for  example,  in  an  electric  motor  drive  or  in  motor- 
truck service. 

183.  Roller  Chains. — A  typical  roller  chain  is  shown  in  Fig. 
105.  This  type  of  chain  is  used  to  some  extent  in  motor-vehicle 
service,  especially  on  trucks,  as  well  as  for  general  power  trans- 
mission. Chain  speeds  as  high  as  1,400  feet  per  minute  have 
been  used  successfully  on  light  loads;  but  for  general  use  with 
proper  lubrication  .1,200  feet  per  minute  should  be  the  limit. 
Occasionally  double  roller  chains  are  used  and  if  properly  in- 


ART.  184] 


ROLLER  CHAIN  SPROCKETS 


243 


stalled  they  give  good  service.  In  Table  59  are  given  the  com- 
mercial sizes  and  other  information  pertaining  to  the  roller 
chain  made  by  the  Diamond  Chain  and  Mfg.  Co.  Instead  of 
the  ultimate  strength  of  the  chain,  the  normal  and  maximum 
allowable  loads  are  given.  The  normal  loads  are  based  on  a 
bearing  pressure  of  1,000  pounds  per  square  inch  of  the  projected 
area  of  the  rivet,  while  the  maximum  load  is  approximately  three 
times  the  normal  but  in  no  case  will  it  exceed  one-tenth  of  the 
ultimate  strength  of  the  chain. 

In  arriving  at  the  size  of  a  roller  chain  required  for  a  particular 
duty,  the  various  points  mentioned  in  Art.  182  apply  equally 
well  in  the  present  case. 


FIG.  105. 

184.  Sprockets  for  Roller  Chains. — As  in  the  case  of  block 
chains,  the  sprockets  used  with  high-grade  roller  chains  are  always 
made  with  cut  teeth.  The  forms  given  to  the  teeth  by  the  vari- 
ous manufacturers  of  roller  chains  differ  considerably. 

(a)  Old-style  tooth  form. — In  Fig.  106  is  shown  a  tooth  form 
that  is  faulty  in  that  it  makes  no  provision  for  the  stretching  of 
the  chain  due  to  wear  on  the  pins  or  rivets.  If  the  space  between 
the  teeth  were  made  wider,  as  shown  in  Fig.  108(a),  giving  the 
roller  more  clearance,  the  chain  drive  would  be  satisfactory.  At 
the  present  time  cutters  that  give  a  clearance  approximating  one- 
tenth  of  the  radius  of  the  chain  roller  are  used  in  the  manufacture 
of  sprockets.  As  the  chain  runs  on  or  off  the  sprocket,  the  curve 
described  by  the  roller  is  an  involute  of  the  pitch  circle,  from 
which  it  would  appear  that  the  face  of  the  tooth  should  be  made 
an  involute.  This,  however,  is  not  done  as  the  face  of  the  tooth 
is  generally  made  an  arc  of  a  circle  a  trifle  inside  of  the  involute 


244 


TABLE  OF  DIAMOND  ROLLER  CHAINS        [CHAP.  X 


in  order  that  the  roller  will  have  no  contact  with  the  tooth  on 
entering  or  leaving  the  sprocket.  The  length  of  the  addendum 
of  the  tooth  is  arbitrarily  taken  as  one-half  of  the  diameter  of 
the  roller.  The  pitch  diameter  of  the  sprocket  is  obtained  by 
the  use  of  formula  (275)  derived  for  the  common  detachable 
chain  in  Art.  178. 

TABLE  59. — DIAMOND  ROLLER  CHAINS 


Chain 
No. 

Pitch 

Roller 

Diam.  of 
rivet 

Weight 
per  foot 

Allowable  load 

Remarks 

Length 

Diam. 

Normal 

Maximum 

75 

K 

H 
K« 

K 

0.306 

JK* 

0.280 
0.300 
0.320 

44 
55 
65 

120 

Single  roller 

147-149 

H 

K 
« 

0.4 

0.200 

0.475 
0.619 

83 
108 

250 
325 

Single  roller 

151 

IK 

K 
H 
K 

H 

0.312 

1.580 
1.690 
1.800 

253 
292 
331 

760 
877 
994 

Single  roller 

153 

H 

X* 
H 
K 
H 
K 

0.469 

0.220 

0.710 
0.760 
0.860 
0.960 

106 
120 
147 
175 

317 
359 
442 
500 

Single  roller 

1.450 

295 

750 

Double  roller 

154 

l 

M 
% 
K 
H 

H 

0.312 

1.680 
1.810 
1.940 

253 
292 
331 

760 
877 
994 

Single  roller 

3.290 

585 

1,700 

Double  roller 

155 

1 

H 

K 
H 

K 

He 

0.281 

1.070 
1.170 
1.270 

176 
211 
246 

527 
632 
738 

Single  roller 

1.840 

422 

1,200 

Double  roller 

157 

1M 

H 

H 

0.375 

2.410 
2.740 

396 
492 

1,189 
1,476 

Single  roller 

160 

IK 

H 
H 

y± 

0.375 

2.540 
2.690 
2.990 

350 
396 
492 

1,049 
1,189 
1,476 

Single  roller 

4.850 

793 

2,400 

Double  roller 

162 

IK 

H 

i 

H 

0.437 

3.890 
4.150 

520 
629 

1,560 

1,888 

Single  roller 

164 

IX 

i 

1 

0.500 

4.960 

720 

2,160 

Single  roller 

8.750 

1,440 

4,000 

Double  roller 

168 

2 

IK 

IK 

0.5625 

6.320 

975 

2,925 

Single  roller 

11.560 

2,231 

6,000 

Double  roller 

(6)  Diamond  tooth  form. — In  Fig.  107  is  shown  the  method  used 
by  the  Diamond  Chain  and  Mfg.  Co.  for  laying  out  their  latest 
type  of  sprocket.  The  information  given  in  the  figure  as  well  as 
the  formula  below  were  kindly  furnished  by  Mr.  G.  M.  Bartlett, 


ART.  184] 


ROLLER  CHAIN  SPROCKETS 


245 


mechanical  engineer  for  the  firm.     In  the  following  formulas  p 
represents  the  pitch  of  the  chain  as  shown  in  Fig.  105. 


FIG.  106. 


a  =  chain  width  —  0.045  p 

b  =  0.545  p 

c  =  0.3  p 

d  =  diameter  of  roller 


(284) 


The  angle  of  pressure  between  the  roller  and  the  tooth  is  20 
degrees,  as  shown  in  the  figure. 


r-o  -*. 


FIG.  107. 


(c)  Renold  tooth  form. — Another  recent  design  of  sprocket  tooth 
form  is  illustrated  in  Fig.  108(6).  It  represents  the  results  of 
many  years  of  experience  with  roller  chains  as  well  as  several 
years  of  special  research  work  by  Mr.  Hans  Renold,  a  prominent 


246 


LENGTH  OF  ROLLER  CHAIN 


[CHAP.  X 


English  chain  manufacturer.  The  results  of  his  work  were  pre- 
sented before  the  American  roller  chain  manufacturers  in  the 
spring  of  1914.  The  form  of  the  tooth,  which  is  not  protected 
by  patents,  has  a  distinct  advantage  over  the  older  forms  still 
used  by  some  chain  makers,  in  that  the  stretch  of  the  chain  is 
taken  care  of  by  the  rollers  rising  on  the  tooth  flanks.  The  tooth 
is  thus  prevented  from  wearing  into  a  hook  form  and  a  smooth- 
running  transmission  is  insured. 

The  space  between  the  teeth  is  made  an  arc  of  a  circle  having  a 
radius  equal  to  the  diameter  of  the  roller  or  a  few  thousands  of  an 


FIG.  108. 

inch  larger.  The  straight  lines  forming  the  teeth  are  tangent  to 
this  arc  and  make  an  angle  of  60  degrees  with  each  other  as  shown 
in  the  figure.  The  face  of  the  tooth  is  relieved  near  the  top  by  a 
circular  arc.  The  height  of  the  tooth  thus  formed  is  greater  than 
that  used  with  other  tooth  designs. 

185.  Length  of  Roller  Chain. — It  is  evident  that  a  chain  cannot 
have  a  fractional  number  of  pitches  or  links ;  hence  in  all  cases  the 
next  whole  number  above  the  calculated  number  must  be  selected, 
and  if  the  distance  between  the  centers  of  the  driving  and 
driven  sprocket  will  permit  a  slight  change,  the  number  chosen 
should  be  an  even  number.  An  odd  number  of  pitches  will 
necessitate  the  use  of  an  offset  link  for  joining  the  ends  of  the 
chain.  The  following  formula  used  by  the  Diamond  Chain  and 


ART.  186]  SILENT  CHAINS  247 

Mfg.  Co.  gives  the  chain  length  in  pitches  and  has  been  found  to 
give  accurate  results: 

Chain  length)  =  2L  +  ^  ^  +  ^  +  0.0257         _ 
in  pitches    j  -^ 

in  which  L  denotes  the  distance  between  the  centers  of  the  two 
sprockets,  and  T\  and  T2  the  number  of  teeth  on  the  large  and 
small  sprocket,  respectively.  If  it  is  desirable  to  determine  the 
length  of  the  chain  in  inches,  merely  multiply  the  pitch  by  the 
chain  length  obtained  from  (285). 

186.  Silent    Chains. — The    best   forms   of   chain   capable   of 
transmitting    power    at    high  speeds  are  those   designated   as 
silent  chain.     An  installation  of  such  a  chain  if  properly  designed 
and  constructed  will  be  just  as  efficient  as  a  gear  drive  for  the 
same  conditions  of  operation.     At  the  present  time  there  are  in 
use  several  designs  of  silent  chain,  having  in  general  the  same 
form  of  link  and  differing  only  in  the  type  of  joint  used.     With 
silent  chains,  the  load  transmitted  is  distributed  equally  between 
all  of  the  sprocket  teeth  in  contact  with  the  chain,  and  is  not 
carried  by  a  single  tooth  as  is  the  case  in  some  of  the  chains  here- 
tofore discussed. 

Silent  chains  are  well  adapted  for  transmitting  power  economic- 
ally at  speeds  of  1,200  to  1,500  feet  per  minute.  The  lower  speed 
holds  for  chains  having  a  pitch  greater  than  one  inch  and  the 
higher  value  for  small  chains.  If  the  speed  is  in  excess  of  1,500 
feet  per  minute,  chains  are  liable  to  be  noisy  unless  they  are 
enclosed  and  run  in  oil.  With  properly  designed  gear  cases 
and  with  the  use  of  good  lubricants,  the  smaller  sizes  of  chains 
may  be  run  at  2,000  feet  per  minute  and  the  larger  sizes  at  1,500. 
It  should  be  borne  in  mind,  however,  that  these  speeds  are 
attained  at  the  cost  of  reduced  life  of  the  chain.  Where  a  positive 
drive  is  essential,  as  in  direct-connected  motor-driven  machinery, 
and  where  the  shafts  are  too  far  apart  for  gearing,  silent  chains 
are  used  extensively.  Chains  transmitting  power  in  dusty  and 
dirty  surroundings  should  always  be  enclosed  in  an  oil-tight  case. 

187.  Coventry  Chain. — The  Coventry  chain  shown  in  Fig.  109 
is  manufactured  in  England,  but  is  used  to  a  considerable  extent 
in  America.     It  consists  of  links  of  special  form  assembled  in 
pairs  and  held  together  by  the  hardened  steel  bushes  b.     Various 
widths  of  chains  are  produced  by  assembling  these  double  links 


248 


WHITNEY  CHAIN 


[CHAP.  X 


alternately  on  hardened  steel  pins;  for  example,  the  chain  shown 
in  Fig.  109  is  called  a  1  X  2  combination.  The  links  themselves 
are  not  hard,  and  their  shape  is  such  that  the  load  is  distributed 
equally  over  all  the  teeth  on  the  sprocket  in  actual  contact  with 


FIG.  109. 


the  chain.     This  action  is  illustrated  in  Fig.  110,  which  also  shows 
the  form  of  tooth  used  on  such  sprockets. 

188.  Whitney  Chain.— The  chain  illustrated  by  Fig.  Ill  is  an 
American  design,  manufactured  by  The  Whitney  Mfg.  Co.  of 


FIG.  110. 

Hartford,  Conn.  The  shape  of  the  links  in  this  chain  is  similar  to 
that  used  on  the  Coventry  chain,  and  hence  the  action  of  the  links 
on  the  sprocket  teeth  is  practically  the  same.  The  individual 
load  links  turn  on  the  outside  of  the  hard  steel  bushes  b  which  are 
fastened  securely  into  the  guide  plates  a.  The  hardened  steel 


ART.  189] 


LINK  BELT  CHAIN 


249 


pins  turning  within  the  bushings  are  forced  into  outside  steel 
plates  shaped  like  the  figure  eight.  The  function  of  the  outside 
plates  is  to  increase  the  tensile  strength  of  the  chain. 


FIG.  111. 

189.  Link  Belt  Chain.— The  Link  Belt  Co.,  after  manufactur- 
ing for  several  years  a  plain  pin-joint  silent  chain  patented  by 
Hans  Renold  of  England,  finally  introduced  the  chain  illustrated 
in  Fig.  112.  The  joint  consists  of  a  case-hardened  steel  pin  hav- 
ing a  bearing  on  two  case-hardened  steel  bushes  b  and  c.  These 


FIG.  112. 

bushes  are  segmental  in  shape  and  are  fitted  into  broached  holes 
in  the  links,  as  shown  in  the  figure.  This  type  of  joint  increases 
the  bearing  area  on  the  pin  over  that  obtained  in  the  original 
Renold  chain  that  had  no  bushes  at  all.  This  chain  is  not  pro- 
vided with  guide  plates,  so  special  provisions  must  be  made  on 
the  sprocket  for  retaining  it. 


250 


MORSE  CHAIN 


[CHAP.  X 


190.  Morse  Chain. — In  the  Morse  chain  shown  in  Fig.  113,  the 
joint  is  of  a  peculiar  construction  in  that  it  introduces  rolling  fric- 
tion in  place  of  the  sliding  friction  common  to  all  the  types  of 
silent  chains  discussed  in  the  preceding  articles.  The  joint  con- 
sists of  two  hardened  steel  pins  6  and  c  anchored  securely  in  their 
respective  ends  of  the  link.  The  pin  b  has  a  plane  surface  against 
which  the  edge  of  the  pin  c  rolls  as  the  chain  runs  on  or  off  the 
sprocket.  The  wear  all  comes  upon  the  two  pins  and  these  may 
be  easily  renewed.  When  the  chain  is  off  the  sprocket  the  load 
upon  the  joints  for  that  part  of  the  chain  between  the  sprockets 


FIG.  113. 

is  taken  by  relatively  flat  surfaces  and  not  by  the  edge  of  the  pin 
c.  It  is  probable  that  the  Morse  chain  will  give  better  service  in 
dusty  places  than  any  other  type  of  silent  chain,  due  to  the  fact 
that  the  rocker  joint  used  requires  less  lubrication  than  the  cylin- 
drical pin  joints. 

191.  Strength  of  Silent  Chains.— The  life  of  a  silent  chain  de- 
fends upon  the  bearing  area  of  the  pins  or  bushings  and  not  so 
much  upon  the  ultimate  strength.  For  minimum  wear  of  the 
chain  and  for  maintained  efficiency,  the  working  load  under 
normal  conditions  approximates  one-thirtieth  of  the  ultimate 
strength,  while  under  severe  fluctuations  of  load  at  the  maximum 
speed  it  is  taken  as  one-fiftieth  of  the  ultimate  strength.  Some 
manufacturers  limit  the  bearing  pressure  on  the  pins  to  650 
pounds  per  square  inch  of  projected  pin  area.  Since  the  strength 


ART.  192]  SILENT  CHAIN  SPROCKETS  251 

of  a  chain  can  be  increased  by  merely  adding  to  its  width,  it  is 
evident  that  for  the  same  load  conditions,  chains  of  different 
pitches  and  widths  may  be  selected;  for  example,  a  1-inch  pitch 
chain  4  inches  wide  and  a  1^-inch  pitch  chain  3  inches  wide  are 
capable  of  transmitting  approximately  the  same  horse  power  at 
the  same  speed.  Experience  dictates  that  the  width  should  range 
from  two  to  six  times  the  pitch. 

The  first  cost  of  narrow  chains  having  a  long  pitch  is  less  than 
wide  ones  of  a  shorter  pitch.  The  longer  pitch  chains  require 
larger  sprockets,  but  are  to  be  preferred  when  the  distance  between 
the  connected  shafts  is  great.  Frequently  it  is  found  desirable 
to  run  two  chains  side  by  side  in  ,order  to  transmit  the  desired 
horse  power. 

In  Table  60  is  given  information  pertaining  to  the  Morse 
chain,  which  will  serve  for  making  the  preliminary  study 
of  a  silent-chain  installation.  This  information  was  kindly  fur- 
nished by  the  Morse  Chain  Co.  of  Ithaca,  N.  Y.  Table  61  con- 
tains useful  data  relating  to  the  Whitney  chain,  while  Table  62 
applies  to  the  Link  Belt  chain. 

192.  Sprockets  for  Silent  Chains. — An  inspection  of  the 
figures  illustrating  the  various  types  of  silent  chains  shows  that 
the  shapes  of  the  individual  links  are  all  about  alike.  The  angle 
included  between  the  working  faces  of  the  link  is  made  60  degrees 
by  all  of  the  manufacturers;  hence  it  follows  that  the  angle  in- 
cluded between  the  flanks  of  alternate  teeth  will  always  be  60 
degrees  irrespective  of  the  number  of  teeth  in  the  sprocket. 
However,  the  angle  included  by  the  flanks  of  the  same  tooth  will 
change,  being  small  for  lower  numbers  of  teeth.  As  this  angle 
decreases  rapidly  for  sprockets  having  small  numbers  of  teeth, 
the  manufacturers  try  to  limit  the  number  of  teeth  in  small 
sprockets  to  15.  Whenever  the  installation  permits,  and  when 
very  quiet  operation  is  desirable,  the  lower  limit  is  placed  at  17. 
Again,  since  the  angle  between  the  flanks  of  the  same  tooth  in- 
creases with  the  number  of  the  teeth  in  the  sprocket,  it  is  found 
necessary  to  limit  the  number  of  teeth  to  about  120  or  130  on 
account  of  the  liability  of  the  chain  to  slide  over  the  teeth.  The 
tooth  form  for  any  particular  make  of  chain  is  determined  best 
by  laying  it  out  to  conform  to  the  dimensions  of  the  links  to  be 
used. 

The  so-called  pitch  diameter  of  the  sprocket  for  silent  chain 


252 


MORSE  CHAIN  DATA 


[CHAP.  X 


FHwJjW  i 


^ 

t~  ^H  ^  O5  £H  CO  O  O  »C  >C 

i-*Wji.w  i  «o  •* 

1-1  '"  «3  C  id 


O  O  »C  >C       O 

•*  TJ<  •*  co  o  o  o  o 

N°  -<li  d  <N 


1C 
Vt       i-H 

>  i  o  i-i 


b-      OOO  iC 

t-  o  o  t--  I-H     QCO 

-*<  CO  !O  IN  <N  iC  O  O  O 


fot^17loit7l°oooio     ooo 

TH'H  Jj  <»  ic  -"-1  T}100  <°  "5  0  0  -I 

-"     "5  d  d  "^        drndd 


Minim 

Desirabl 
Maxi 
Desir 
To  fi 
2 
M 


8s 

aa 


P. 

•sS 


"0*0 


22 
aa 

aa 


Q 

a 

a>     ~S 

II 


"^  c  S    72,j£'%  « 

•la;  ills 
gs1  «&'^ 

'    1585 


I 


-S  |IS 
!s  sig. 


»^.a   .  GTS  u^ 

3  '  u  3  £  o—1  O  O.Q 

of!  °  9  ^nv?  rt  o  " 


1 
s 


ART.  192] 


TABLE  OF  WHITNEY  CHAINS 


253 


having  pin  joints  may  be  determined  by  the  formula  used  for 
roller  chains,  namely 


D  = 


P 


sin  a 


in  which  p  denotes  the  pitch  of  the  chain  and  the  angle  a  is 
equal  to  180  degrees  divided  by  the  number  of  teeth.  Whenever 
possible  an  odd  number  of  teeth  should  be  used  for  the  pinion  so 
that  the  wear  may  be  distributed  more  evenly.  Sprockets  should 
be  made  as  large  as  possible  to  relieve  the  wear  on  the  chain,  as 
in  passing  around  small  sprockets  the  angular  displacements  of 
each  link  on  the  pins  or  bushes  is  greater  than  in  the  case  of  large 
sprockets. 

TABLE  61. — WHITNEY  SILENT  CHAINS 


1 

a 
•3 

JS 
O 

o 

s 

Width 
between 
guide  links 

Weight  per  ft. 

Maximum 
speed, 
ft.  per  min. 

Ultimate 
strength,  Ib. 

d 
d 

JS 

O 

w 

£ 

Width 
between 
guide  links 

Weight  per  ft. 

Maximum 
speed,  ft.  per 
min. 

Ultimate 
strength,  Ib. 

1201 

K 

0.56 

2,800 

1265 

IX 

3.22 

14,400 

1202 

K 

0.74 

3,400 

1266 

2 

3.59 

15,600 

1203 

H 

1 

0.92 

4,000 

1267 

2K 

3.96 

16,800 

1204 

IK 

1.10 

4,600 

1268 

2H 

4.33 

18,000 

1205 

IK 

1.28 

5,200 

1269 

1  97fl 

4.70 
5O7 

19,200 
20  400 

1221 

K 

0.83 

4,900 

l«f  U 

1271 

% 

3K 

.  U/ 

5.44 

21,600 

1222 

2i 

1.08 

5,800 

1272 

3K 

5.81 

22,800 

1223 

1 

1.33 

6,700 

1273 

3^4 

6.18 

24,000 

1224 

K 

IK 

1.58 

7,600 

1274 

4 

6.55 

25,200 

1225 

IK 

1.83 

8,500 

1275 

4K 

6.92 

26,400 

1226 

1% 

2.08 

9,400 

1276 

4K 

7.29 

27,600 

1227 

2 

2  33 

i  o  *^nn 

* 

1281 

3  24 

17,200 

1241 

K 

1.35 

7,100 

1282 

IK 

4.25 

2o!lOO 

1242 

1 

1.66 

8,000 

1283 

2 

5.26 

23,000 

1243 

IK 

1.97 

8,900 

1284 

2K 

6.27 

25,900 

1244 

IK 

2.28 

9,800 

1285 

3 

7.28 

28,800 

1245 

IK 

2.59 

10,700 

1286 

3K 

8.29 

31,700 

1246 

2 

2.90 

11,600 

1287 

4 

9.30 

34,600 

1247 

M 

2K 

3.21 

12,500 

1288 

i 

4K 

10.31 

37,500 

1248 

2K 

3.52 

13,400 

1289 

5 

11.32 

40,400 

1249 

2^ 

3.83 

14,300 

1290 

5K 

12.33 

43,300 

1250 

3 

4.14 

15,200 

1291 

6 

13.34 

46,200 

1251 

3K 

4.45 

16,100 

1292 

8K 

14.35 

49,100 

1252 

3K 

4.76 

17,000 

1293 

7 

15.36 
1007 

52,000 

1261 

K 

1.74 

9,600 

1295 

8 

in  .  of 
17.38 

57,800 

1262 

1 

2.11 

10,800 

1263 

K 

IK 

2.48 

12,000 

1264 

IK 

2.85 

13,200 

254 


TABLE  OF  LINK  BELT  CHAINS 


[CHAP.  X 


TABLE  62. — HORSE  POWER  TRANSMITTED  BY  LINK  BELT  SILENT  CHAIN 


Pitch  of 
chain 

ofl 

11 

Speed  of  chain  in  ft.  per  min. 

500 

600 

700 

800 

900 

1,000 

1,100 

1,200 

1,300 

1,400 

1,500 

K 

0.58 

0.66 

0.72 

0.78 

0.82 

0.88 

0.91 

0.95 

y* 

0.87 

0.98 

1.07 

1.16 

1.22 

1.30 

1.38 

1.42 

i 

1.16 

1.31 

1.43 

1.55 

1.63 

1.73 

1.82 

1.89 

H 

IK 

1.45 

1.64 

1.79 

1.91 

2.04 

2.18 

2.28 

2.36 

IK 

1.74 

1.97 

2.15 

2.30 

2.45 

2.60 

2.73 

2.83 

2 

2.32 

2.62 

2.86 

3.08 

3.27 

3.46 

3.64 

3.78 

3 
H 

3.48 

3.91 

4.28 

4.61 

4.89 

5.22 

5.46 

5.67 

0.84 

0.95 

1.04 

1.11 

1.19 

1.27 

1.33 

1.38 

1.42 

2£ 

1.26 

1.40 

1.56 

1.70 

1.79 

1.91 

1.99 

2.07 

2.13 

1 

1.68 

1.89 

2.08 

2.25 

2.34 

2.54 

2.65 

2.76 

2.84 

>2 

IK 

2.52 

2.91 

3.12 

3.44 

3.57 

3.88 

3.98 

4.14 

4.25 

2 

3.37 

3.82 

4.17 

4.48 

4.77 

5.10 

5.30 

5.52 

5.68 

3 

5.05 

5.73 

6.25 

6.75 

7.15 

7.60 

7.95 

8.29 

8.50 

4 

6.73 

7.64 

8.30 

9.00 

9.53 

10.10 

10  .  60 

11.10 

11.30 

1 

2.22 

2.51 

2.74 

2.96 

3.15 

3.33 

3.50 

3.64 

3.75 

1H 

2.77 

3.15 

3.41 

3.71 

3.93 

4.18 

4.37 

4.54 

4.70 

IK 

3.33 

3.76 

4.12 

4.43 

4.72 

5.00 

5.25 

5.45 

5.62 

6A 

2 

4.43 

5.02 

5.47 

5.91 

6.30 

6.67 

7.00 

7.28 

7.50 

3 

6.65 

7.52 

8.22 

8.88 

9.45 

10.00 

10.50 

10.90 

11.20 

4 

8.86 

10.00 

10.90 

11.80 

12.60 

13.30 

14.00 

14.50 

15.00 

6 

1 

13.30 

15.00 

16.40 

17.70 

18.90 

20.00 

21.00 

21.80 

22.50 

2.85 

3.22 

3.51 

3.78 

4.05 

4.37 

4.48 

4.65 

4.82 

IK 

3.56 

3.98 

4.39 

4.70 

5.06 

5.30 

5.60 

5.78 

6.02 

IK 

4.27 

4.85 

5.27 

5.67 

6.10 

6.40 

6.72 

6.98 

7.23 

2 

5.68 

6.42 

7.03 

7.56 

8.10 

8.55 

8.95 

9.31 

9.63 

H 

3 

8.55 

9.63 

10.50 

11.40 

12.10 

12.80 

13.40 

14.00 

14.50 

4 

11.40 

12.80 

14.00 

15.10 

16.30 

17.30 

17.90 

18.60 

19.30 

5 

14.20 

16.10 

17.60 

18.90 

20.30 

21.30 

22.40 

23.30 

24.10 

6 

17.10 

19.30 

21.10 

22.80 

24.30 

25.70 

26.80 

27.90 

28.90 

2 

7.00 

7.91 

8.65 

8.33 

10.00 

10.50 

10.90 

11.40 

11.80 

2K 

9.00 

10.10 

11.10 

12.00 

12.90 

13.50 

14.10 

14.70 

15.20 

3 

11.00 

12.40 

13.60 

14.60 

15.70 

16.50 

17.20 

18.00 

18.60 

1 

4 

15.00 

16.90 

18.60 

20.00 

21.50 

22.50 

23.50 

24.60 

25.40 

5 

19.00 

21.50 

23.50 

25.20 

27.20 

28.70 

29.70 

31.10 

32.10 

6 

23.00 

26.00 

28.50 

30.50 

32.90 

34.50 

36.00 

37.60 

38.90 

8 

31.00 

34.90 

38.40 

41.20 

44.30 

46.30 

48.50 

50.70 

52.40 

2 

9.70 

11.00 

11.90 

13.00 

13.80 

14.60 

15.30 

15.90 

16.40 

16.7 

3 

15.30 

17.30 

18.70 

20.30 

21.70 

22.90 

24  .  20 

25.00 

25.70 

26.5 

4 

20.80 

23.50 

25.50 

27.60 

29.60 

31.20 

32.60 

34.10 

35.10 

36.2 

1H 

5 

26.30 

29.80 

32.30 

35.10 

37.50 

39.70 

41.60 

43.20 

44.50 

45.8 

6 

31.80 

36.20 

39.10 

42.70 

45.30 

48.20 

50.30 

52.20 

53.80 

55.5 

8 

42.80 

48.50 

52.70 

57.20 

61.20 

64.00 

67.80 

70.30 

72.50 

74.6 

10 

54.10 

61.30 

66.50 

72.20 

77.10 

81.20 

85.60 

88.70 

91.40 

94.1 

ART.  192]  TABLE  OF  LINK  BELT  CHAINS  255 

TABLE  62. — HORSE  POWER  TRANSMITTED  BY  LINK  BELT  SILENT  CHAIN  (Cont.) 


0 

Speed  of  chain  in  ft.  per  min. 

^.s 

X3-S 

£l 

••5  j= 

*' 

500 

600 

700 

800 

900 

1,000 

1,100 

1,200 

1,300 

1,400 

1,500 

3 

20.10 

22.70 

24.70 

26.90 

28.70 

30.30 

31.80 

33.00 

34.00 

35. 

35.7 

4 

27.50 

31.10 

33.70 

36.60 

39.10 

41.20 

43.40 

45.00 

46.40 

48.0 

48.7 

5 

34.80 

39.30 

42.70 

46.30 

49.50 

52.30 

55.00 

57.00 

58.70 

60.7 

61.6 

\y^ 

6 

42.20 

47.60 

51.80 

56.30 

60.00 

63.40 

66.50 

69.00 

71.10 

73.5 

74.7 

8 

56.70 

64.20 

69.70 

75.70 

81.00 

85.20 

89.70 

93.00 

95.80 

99.0 

101.0 

10 

71.40 

80.70 

87.70 

95.20 

102.00 

107  .  00 

113.00 

117.00 

121.00 

124.0 

127.0 

12 

86.00 

97.30 

106.00 

115.00 

123.00 

129.00 

136.00 

141.00 

145.00 

150.0 

152.0 

6 

56.10 

63.50 

69.00 

75.00 

80.00 

84.30 

88.80 

92.00 

94.80 

97.5 

99.6 

8 

75.70 

85.60 

93.00 

101.00 

108.00 

114.00 

120.00 

124.00 

128.00 

131.0 

134.0 

2 

10 

95.20 

107.00 

117.00 

126.00 

136.00 

143.00 

151.00 

156.00 

161.00 

165.0 

169.0 

12 

114.00 

129.00 

141.00 

153.00 

164.00 

172.00 

182.00 

188.00 

194.00 

199.0 

204.0 

14 

134.00 

152.00 

165.00 

179.00 

191.00 

201.  OQ 

212.00 

220.00 

227.00 

233.0 

240.0 

16 

154.00 

174.00 

189.00 

205.00 

220.00 

231.00 

243.00 

252.00 

260.00 

267.0 

273.0 

6 

73.00 

82.70 

90.00 

98.00 

104.00 

110.00 

116.00 

120.00 

124.00 

127.0 

130.0 

8 

100.00 

113.00 

123.00 

133.00 

143.00 

150.00 

158.00 

164.00 

169.00 

174.0 

178.0 

2>2 

10 

126.00 

143.00 

155.00 

168.00 

180.00 

190.00 

200.00 

207.00 

213.00 

220.0 

224.0 

12 

153.00 

173.00 

188.00 

204.00 

218.00 

230.00 

242.00 

251.00 

259  .  00 

266.0 

272.0 

14 

179.00 

204  .  00 

220  .  00 

240.00 

255.00 

270.00 

284.00 

294  .  00 

303.00 

313.0 

318.0 

16 

206.00 

235.00 

253.00274.00 

294.00 

310.00 

326.00 

338.00 

348  .  00 

359.0 

365.0 

I 

I 

I 

The  several  makes  of  so-called  silent  chains  require  different 
types  of  sprockets  in  order  to  keep  the  chain  from  running  off,  as 


FIG.  114 


may  be  noticed  by  consulting  Figs.  114  and  115.  The  so-called 
outside-guided  chain  shown  in  Figs.  109  and  111  require 
plain  sprockets,  since  the  guide  links  prevent  it  from  running  off. 
The  Link  Belt  chain,  having  no  guide  links,  depends  upon  flanged 


256 


SPRING-CUSHIONED  SPROCKETS 


[CHAP.  X 


sprockets  of  one  form  or  another.  One  design  of  such  a  sprocket, 
as  used  by  the  Link  Belt  Co.,  is  shown  in  Fig.  115,  and  in  Table  63 
are  given  some  general  proportions  pertaining  thereto.  The 
Morse  chain  is  always  provided  with  central  guide  links;  hence, 
the  sprocket  teeth  are  provided  with  one  or  more  central  grooves 
in  which  the  guide  plates  run.  A  design  of  this  description  is 
shown  in  Fig.  117. 


FIG.  115. 
TABLE   63. — GENERAL  PROPORTIONS  OF  LINK  BELT  SPROCKETS 


Dimensions 

Chain 

pitch 

a 

b 

c 

e 

7 

H 

KG 

y2 

H* 

•3  fc,d 

y* 

0.2 

KG 

H 

all 

5/8 

0.25 

i  i/fg 

CO    ». 

0.3 

H 

2e  +/ 

%2 

'fl  ^  '1 

i 

0.4 

'53  ^  'o 

IM 

0.5 

1% 

Ha 

1  ^^i 

1% 

0.6 

1% 

^  a  a 

2 

0.85 

2H6 

KG 

.22^ 

2^ 

1.25 

21/2 

KG 

x-  - 

193.  Spring-cushioned  Sprockets. — In  a  power  transmission 
subjected  to  shocks  due  to  intermittent  and  irregular  loads,  it  is 


ART.  193] 


SPRING-CUSHIONED  SPROCKETS 


257 


considered  good  practice  to  use  a  form  of  sprocket  that  is  capable, 
of  absorbing  these  shocks  thereby  relieving  the  chain.  In  general, 
such  a  device  (see  Fig.  116  or  117)  consists  of  an  inner  hub  a 
keyed  to  the  shaft,  and  upon  this  hub  is  mounted  the  sprocket 
rim  e.  Between  the  lugs  6,  cast  integral  with  a,  and  the  lugs 
d  on  the  inside  of  the  rim  e  are  placed  the  compression  springs  c, 
through  which  the  driving  load  must  be  transmitted.  The  design 
^hown  in  Fig.  116  is  furnished .  with  a  cover  plate  /  to  make  it 
dustproof,  and  is  representative  of  the  practice  of  the  Link  Belt 


FIG.   116. 

Co.  The  Morse  Chain  Co.  spring-cushion  sprocket,  shown  in 
Fig.  117,  is  also  dustproof  but  the  split-rim  construction  is  used. 

It  is  suggested  that  spring-cushioned  sprockets  are  well  adapted 
to  such  service  as  is  met  with  in  driving  air  compressors,  pumps, 
metal  planers  and  shapers,  and  punching  and  shearing  machinery; 
however,  they  are  not  used  to  any  extent  in  such  places,  no  doubt 
due  to  the  additional  cost. 

Whenever  two  chains  are  used  side  by  side  to  transmit  a  given 
horse  power,  a  "compensating  sprocket"  should  be  used  unless  the 
transmission  is  horizontal  and  the  distance  between  the  shafts  is 
considerable  so  that  quite  a  little  weight  of  chain  is  between  the 
sprockets.  A  compensating  sprocket  may  be  made  by  mounting 


258 


REFERENCES 


[CHAP.  X 


two  spring-cushioned  sprockets  side  by  side  on  one  central  hub, 
thus  dividing  the  load  equally  between  the  chains. 

In  the  design  of  cushioned  sprockets  for  intermittent  work, 
for  example,  driving  reciprocating  pumps  not  subjected  to  a 
water-hammer  or  excessive  overloading,  the  compressive  load  on 
the  springs  should  be  based  on  a  chain  load  two  and  one-half  to 
three  times  the  actual  load.  In  installations  where  water-ham- 
mers on  pumps,  or  other  heavy  additional  loads,  would  come  upon 
the  springs,  the  latter  should  be  designed  for  loads  from  four  to 
five  times  the  actual  load  on  the  chain. 


FIG.  117. 


References 

Elements  of  Machine  Design,  by  W.  C.  UNWIN. 

Machine  Design,  Construction  and  Drawing,  by  H.  J.  SPOONER. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Mechanical  Engineers'  Handbook,  by  L.  S.  MARKS. 

The  Strength  of  Chain  Links,  Bull.  No.  18,  Univ.  of  Illinois  Experiment 
Station. 

A  Silent  Chain  Gear,  Trans.  A.  S.  M.  E.,  vol.  23,  p.  373. 

Roller  Chain  Power  Transmission  and  Construction  of  Sprockets,  Mchy., 
vol.  11,  p.  287. 

Chart  for  Chain  Drives,  Amer.  Mach.,  vol.  37,  p.  854. 

Calculations  for  Roller  Chain  Drives,  Mchy.,  vol.  20,  p.  567. 

The  Manufacture  of  Chain,  Mchy.,  vol.  21,  pp.  719  and  817. 

Roller  and  Silent  Chain,  Trans.  Soc.  of  Auto.  Engrs.,  vol.  5,  p.  390. 

Silent  Chain  Power  Transmission,  Paper  before  the  Assoc.  of  Iron  and 
Steel  Elect.  Engrs.,  September,  1914. 

The  Transmission  of  Power  by  Chains,  Birmingham  Assoc.  of  Mech. 
Engrs.,  November,  1914. 

Link  Belt  Silent  Chain,  Data  Book,  No.  125,  Link-Belt  Co. 

Power  Chains  and  Sprockets,  Diamond  Chain  and  Mfg.  Co. 

Diamond  Tooth  Form  for  Roller  Chain  Sprockets,  Diamond  Chain  and 
Mfg.  Co. 


CHAPTER  XI 
FRICTION  GEARING 

Friction  gearing  is  employed  when  the  positiveness  of  relative 
motion  is  either  unnecessary  or  not  essential.  The  wheels  de- 
pend for  their  driving  value  upon  the  coefficient  of  friction  of  the 
composition  wheel  against  its  iron  mate,  and  their  actual  driving 
capacity  becomes  a  function  of  the  pressure  with  which  they  are 
held  in  contact.  This  pressure  is  limited  by  the  ability  of  the 
composition  surface  to  endure  it  without  injury.  The  composi- 
tion wheel  should  never  be  used  as  the  driven  member  of  a  pair  of 
wheels,  since,  being  of  a  softer  material,  its  surface  would  be 
injured  and  eventually  ruined  by  the  occasional  rotation  of  the 
iron  wheel  against  it  under  pressure  before  starting  it  from  rest, 
or  after  an  excessive  load  has  brought  it  to  a  standstill.  Friction 
gearing  may  be  used  for  transmitting  power  between  shafts  that 
are  parallel  or  between  those  that  intersect. 

194.  Experimental  Results. — Several  years  ago  an  extended 
series  of  experiments  on  friction  gearing  was  made  at  the  labora- 
tory of  Purdue  University,  the  results  of  which  were  reported  by 
Prof.  Goss  in  a  paper  before  the  American  Society  of  Mechan- 
ical Engineers.  These  experiments  were  made  upon  compressed 
strawboard  driving  wheels  approximately  6,  8,  12  and  16  inches 
in  diameter  in  contact  with  a  turned  cast-iron  follower  16  inches 
in  diameter.  The  pressures  per  inch  of  face  varied  from  75  to 
more  than  400  pounds,  and  the  tangential  velocity  from  400  to 
2,800  feet  per  minute.  The  following  are  some  of  the  conclusions 
derived  from  these  tests : 

(a)  Slippage  increases  gradually  with  the  load  up  to  3  per  cent., 
and  when  it  exceeds  this  value  it  is  liable  to  increase  very  suddenly 
to  100  per  cent.,  or  in  other  words,  motion  ceases. 

(b)  The  coefficient  of  friction  varies  with  the  slip,  and  becomes 
a  maximum  when  the  slip  lies  between  2  and  6  per  cent. 

(c)  The  coefficient  of  friction  seems  to  be  constant  for  all  pres- 
sures up  to  a  limit  lying  between  150  and  200  pounds  per  inch  of 
face,  but  decreases  as  the  pressure  increases. 

259 


260 


COEFFICIENTS  OF  FRICTION 


[CHAP.  XI 


(d)  The  coefficient  of  friction  is  not  affected  by  variations  in 
the  tangential  velocity  between  the  limits  400  and  2,800  feet  per 
minute. 

(e)  The  coefficient  of  friction  for  the  6-inch  wheel  was  about 
10  per  cent,  less  than  for  the  others. 

(/)  A  coefficient  of  friction  of  20  per  cent,  is  readily  obtained 
with  wheels  8  inches  in  diameter  and  larger. 

In  December,  1907,  Prof.  Goss  presented  before  the  American 
Society  of  Mechanical  Engineers,  a  second  paper  on  the  subject  of 
friction  drives,  in  which  he  reported  the  results  of  another  exten- 
sive series  of  tests.  The  values  of  the  coefficient  of  friction  and 
permissible  working  pressure  per  inch  of  face  for  the  various 
materials  experimented  with  are  given  in  Table  64.  Pressures 

TABLE  64. — EXPERIMENTAL  DATA  PERTAINING  TO  FRICTION  GEARING 


Material 

Coefficient  of  friction- 
working  values 

Safe 
working 
pressure 

Cast 
iron 

Alumin- 
num 

Type 
metal 

Leather 

0.135 
0.150 
0.150 
0.210 
0.255 
0.309 
0.330 

0.216 

0.246 

150 
150 
240 
50 
150 
240 
140 

Wood  

Tarred  fiber  
Cork  composition.  . 
Straw  fiber     

0.183 

0.165 

0.273 
0.297 
0.318 

0.186 
0.183 
0.309 

Leather  fiber  

Sulphite  fiber  

exceeding  150  pounds  per  inch  of  face  may  be  used  providing 
the  conditions  under  which  the  wheels  are  working  are  known 
definitely,  or  where  experience  has  proven  their  use  permissible. 
Several  manufacturers  now  make  wheels  that  allow  the  use  of 
working  pressures  of  250  pounds  or  more. 


SPUR-FRICTION  GEARING 

195.  Plain  Spur  Frictions. — The  simplest  form  of  friction  gear- 
ing consists  of  two  plain  cylindrical  wheels  held  in  contact  with 
each  other  by  properly  constructed  bearings.  Such  wheels, 
shown  in  Fig.  118,  are  known  as  spur  frictions.  To  determine 
the  least  pressure  that  must  be  applied  at  the  line  of  contact  in 
order  that  the  gears  may  transmit  a  given  horse  power,  the  follow- 
ing method  may  be  used : 


ART.  196] 


SPUR-FRICTION  GEARING 


261 


Let  H  =  the  horse  power  transmitted. 

V  =  the  mean  velocity  of  the  gears  in  feet  per  minute. 

/  =  face  of  the  gears. 

p  =  permissible  pressure  per  inch  of  face. 

H  =  coefficient  of  friction. 

Evidently,  the  total  radial  pressure  between  the  two  wheels  at 
the  line  of  contact  is  fp,  and  the  tangential  force  due  to  this 
pressure  is  pfp.  Now  this 
force  must  at  least  equal 
the  tangential  resistance  or 


T  = 


(286)    tt 


Therefore,  the  least  pres- 
sure required  between  the 
two  spur  frictions,  so  that 
H  horse  power  may  be 
transmitted  is 


FlG   118> 


fp  = 


33,000  H 


(287) 


196.  Applications  of  Spur  Frictions. — Plain  spur-friction  gear- 
ing is  used  for  driving  light  power  hoists,  coal  screens,  gravel 
washers,  and  various  forms  of  driers.  Another  useful  and  interest- 
ing application  of  spur  frictions  is  found  in  friction-board  drop 
hammers  used  in  the  production  of  all  kinds  of  drop  forgings. 
Two  designs,  differing  somewhat  in  the  method  of  driving  the 
friction  rolls,  are  shown  in  Figs.  119  and  120.  The  methods  of 
operation  and  control  of  the  hammer  are  similar  in  the  two  de- 
signs. In  Fig.  119,  the  friction  rolls  b  and  c  are  keyed  rigidly  to 
their  respective  driving  shafts  d  and  e  and  may  be  brought  into 
contact  with  the  board  a,  at  the  lower  end  of  which  is  fastened 
the  ram.  It  will  be  noticed  that  the  friction  rolls  are  brought 
into  contact  with  the  board  a  by  rotating  the  eccentric  bearings  in 
which  the  driving  shafts  are  supported.  The  bearings  are  ro- 
tated slightly  by  the  rod/,  which  in  turn  is  tripped  by  the  descend- 
ing hammer.  The  ram  and  the  various  operating  accessories  are 
not  included  in  the  figure. 

The  function  of  the  friction  rolls  6  and  c  is  to  return  the  ram  to 
its  initial  position  after  a  blow  has  been  struck.  As  soon  as  the 
ram  returns  to  its  initial  position,  it  lifts  the  rod  /  by  means  of  a 


262 


ANALYSIS  OF  A  DROP  HAMMER 


[ART.  197 


suitable  mechanism,  and  consequently  the  friction  board  will 
again  drop  unless  it  is  held  by  the  pawls  g  and  h.  These  pawls 
are  controlled  by  the  operator  through  a  treadle. 

The  design  shown  in  Fig.  119  is  that  used  by  the  Billings  and 
Spencer  Co.,  and  differs  from  the  other  in  that  both  shafts  d  and 
e  are  mounted  on  eccentric  bearings,  each  shaft  being  driven  by  a 
belt  and  pulley.  Fig.  120  shows  the  general  details  of  the  design 

used  by  the  Toledo 
Machine  and  Tool 
Co.  The  driving  pul- 
leys are  keyed  to  the 
shaft  e,  which  has 
mounted  upon  it  the 
roll  c  and  a  spur  gear 
m.  The  latter 
meshes  with  the  gear 
n  which  is  fastened 
to  the  roll  6,  both 
being  mounted  with 
a  running  fit  on  the 
shaft  d.  The  shaft  d 
is  supported  on  ec- 
centric bearings  by 
means  of  which  the 
two  rolls  are  brought 
in  contact  with  the 
board  a. 

In  some  designs  of 
drop  hammers,  the 
teeth  on  the  gears  m 
and  n  are  made  of  the 
buttressed  type,  since  they  transmit  power  in  only  one  direction 
and  at  the  same  time  are  subjected  to  a  considerable  shock. 

197.  Analysis  of  a  Drop  Hammer. — The  total  lifting  force  T 
exerted  on  the  friction  board  by  the  driving  rolls  must  exceed  the 
weight  Q  of  the  ram  so  that  it  is  possible  to  accelerate  the  latter 
at  the  beginning  of  the  hoisting  period. 

Let     ti  =  number  of  seconds  required  to  accelerate  the  ram. 
tz  =  number  of  seconds  during  which  the  ram  moves 
upward  at  constant  velocity. 


FIG.  119. 


ART.  197] 


ANALYSIS  OF  A  DROP  HAMMER 


263 


t3  =  number  of  seconds  required  to  bring  the  ram  to 

rest  after  releasing  the  rolls. 
v  =  maximum  velocity  of  ram  during  hoisting  period. 

The  hoisting  period  is  really  made  up  of  three  separate  periods, 
namely:  (1)  the  period  during  which  the  ram  is  accelerated;  (2) 
the  constant-speed  period;  (3)  the  period  immediately  following 
the  releasing  of  the  driving  rolls,  during  which  the  ram  gives  up 
its  kinetic  energy. 

Using  the  above  notation  and  assuming  uniformly  accelerated 
motion,  the  distance  h 
travelled  by  the  ram  in 
its  upward  travel  is 
given  by  the  following 
expression : 

A -§  +  *»+£  (288) 

z  Ag 

Denoting  the  ratio  of  T 
to  Q  by  the  symbol  c  and 
disregarding  the  fric- 
tional  resistances,  it  is 
evident  that  the  acceler- 
ating force  is 

Q(c  -!)  =  !£' 


from  which 


(289) 


0(c- 

Substituting     (289)     in 
(288),  and  simplifying, 

h  =  ^- 
from  which 


FIG.  120. 


_  h  _    v  r     c 
~  v  ~  2~gl^ 


(290) 


The  total  number  of  seconds  required  for  the  hoisting  period  is 

(291) 


264  GROOVED  SPUR  FRICTIONS  [CHAP.  XI 

The  number  of  seconds  required  by  the  ram  to  fall  through  the 
distance  h  is 

(292) 

Hence  the  time  required  for  a  complete  cycle  may  be  readily 
determined. 

During  the  accelerating  period,  the  work  W\  expended  by  the 
friction  rolls  upon  the  board  of  the  ram  is 

*'  =  (293) 


and  during  this  same  period,  the  useful  work  done  is 

W  -  2^1) 
Hence  the  lost  work  is 


The  work  W  represents  the  loss  due  to  slippage  which  will 
tend  to  produce  excessive  temperatures,  thereby  charring  the 
board  of  the  ram;  hence  its  magnitude  must  be  kept  down  by 
using  a  speed  v  that  is  not  too  high,  and  by  making  c  relatively 
large.  In  actual  hammers,  c  varies  from  1.2  to  2. 

The  total  lifting  force  T  is  produced  by  the  pressure  of  the  rolls 
upon  the  board  and  is  given  by  the  relation 

T  =  2MP,  (296) 

in  which  P  denotes  the  normal  pressure  between  each  roll  and 
the  board  and  /x  the  coefficient  of  friction,  which  may  be  assumed 
to  vary  from  0.25  to  0.35. 

198.  Grooved  Spur  Frictions.  —  In  the  case  of  plain  spur  fric- 
tions, the  pressures  upon  the  shafts  are  excessive  for  large  powers, 
thus  causing  a  considerable  loss  of  power  due  to  the  journal  fric- 
tion. To  decrease  this  loss  of  power  by  decreasing  the  pressure 
upon  the  shafts,  a  form  of  gearing  known  as  grooved  spur  fric- 
tions is  used.  Fig.  121  (a)  shows  how  such  gears  are  formed.  It 
is  desired  to  determine  the  relation  between  the  horse  power 
transmitted  and  the  total  radial  pressure  between  the  frictions. 

Let  P  =  radial  thrust  upon  one  projection  or  groove. 

R  =  total  reaction  on  each  side  of  projection  or  groove. 


ART.  198] 


GROOVED  SPUR  FRICTIONS 


265 


T  =  tangential  resistance  on  each  projection  or  groove. 
n  =  number  of  projections  or  grooves  in  contact. 
2  a  =  angle  of  the  grooves. 

In  Fig.  121(6)  are  shown  the  various  forces  acting  upon  one  of 
the   projections.     From   the  force 
triangle  ABC  it  follows  that 


P  =  2R  sin  (a  +  <p), 


(297) 
of 


in  which   <p  denotes  the  angle 
friction  as  shown  in  the  figure. 

In  any  type  of  friction  gearing 
the  tangential  resistance  T  is 
equivalent  to  the  coefficient  of  fric- 
tion multiplied  by  the  total  normal 
pressure  at  the  line  of  contact,  hence 
for  the  case  under  discussion 


T  = 


33,000  H 
nV 


(298) 


in  which  H  and  V  have  the  same 
meaning  as  in  Art.  195. 

Eliminating  the  factor  2R  by 
combining  (297)  and  (298),  the 
least  total  pressure  nP  between  the 
two  grooved  friction  gears  is  given 
by  the  following  expression: 

nP  = 


33,000  H 


(sin  a.  -{-  n  cos  a)    (299) 


From  an  inspection  of  Fig.  121 
(a),  it  is  evident  that  along  the  lines 
of  contact  between  the  two  gears, 
the  so-called  pitch  point  is  the  only 
one  at  which  the  two  gears  have 
the  same  peripheral  speed.  At 
all  other  points  there  is  a  difference 

in  speed  between  the  gears,  and  hence  there  must  be  slippage,  as 
a  result  of  which  excessive  wear  might  be  expected.  In  order 
to  make  this  difference  in  speed  small  and  at  the  same  time  de- 
crease the  resultant  wear,  the  projections  must  be  made  com- 


266 


BEVEL-FRICTION  GEARING 


[CHAP.  XI 


paratively  short.  Furthermore,  the  normal  pressure  per  inch  of 
side  of  groove  or  projection  should  not,  according  to  Bach,  ex- 
ceed 3,200  pounds.  When  a  considerable  number  of  grooves 
are  used,  it  is  necessary  that  they  be  machined  very  accurately 
or  excessive  wear  due  to  high  contact  pressure  will  result.  The 
angle  2  a  of  the  grooves  varies  from  30  to  40  degrees. 


BEVEL-FRICTION  GEARING 

Bevel  frictions  are  used  when  it  is  desired  to  transmit  power 
by  means  of  shafts  that  intersect.  Such  gears  are  shown  in 

Fig.  122.  Referring  to  Fig. 
122,  the  gear  marked  2  is 
keyed  rigidly  to  its  shaft 
while  the  gear  1  is  splined 
to  its  shaft.  By  means  of 
a  specially  designed  thrust 
bearing  operated  by  a  lever 
or  other  means,  the  bevel 
gear  1  is  brought  into  con- 
tact with  gear  2  and  held 
FIG.  122.  there  under  pressure.  In 

designing     a     bevel-friction 

transmission,  both  the  starting  and  running  conditions  should  be 
investigated. 

199.  Starting  Condition. — In  the  following  analysis  it  is 
assumed  that  the  transmission  is  to  be  started  under  full  load,  a 
condition  met  with  frequently  in  connection  with  hoisting 
machinery.  At  the  instant  of  starting,  due  to  the  relative  motion 
between  the  surfaces  in  contact,  the  reaction  R  instead  of  being 
normal  is  inclined  away  from  the  normal  by  the  angle  of  friction 
<f>,  as  shown  in  Fig.  122. 

As  in  the  preceding  cases,  the  tangential  force  that  can  be  trans- 
mitted by  the  two  gears  is  equal  to  the  product  of  the  total  normal 
pressure  and  the  coefficient  of  friction;  thus 

T  =  nR  cos  <f>  =  33,000  ~  (300) 


From  the  geometry  of  the  figure  it  is  evident  that 

Pi  P* 


R 


sin  (a  -\- 


cos  (a 


(301) 


ART.  200]  BEVEL-FRICTION  GEARING  267 

Combining  (300)  and  (301),  the  expressions  for  the  least  axial 
thrusts  that  come  upon  the  gears  are  as  follows: 


Pi  =  (sin  a  +  /*  cos  a)  (302) 

_  ^  gin  a)  (303) 


200.  Running  Condition.  —  After  the  transmission  gets  up  to 
speed,  the  relative  motion  between  the  gears  along  the  line  of  con- 
tact ceases;  hence  the  reaction  between  the  two  surfaces  in  con- 
tact is  normal.  Calling  this  reaction  Rf,  we  have  the  relations 

T'  =  »R'  =  33,000  ~ 
p'  P' 

,  f\  *2 


sin  a        cos  a 

Combining  these  equations  and  solving  for  P[  and  P'z  we  obtain 
the  following: 

as  nnn  77  sin  a 

(304) 


(305) 


The  expressions  just  derived  may  be  put  in  slightly  different 
form  by  substituting  for  sin  a  and  cos  a  their  equivalents  in  terms 
of  the  diameters  DI  and  D2  of  the  gears.  The  resulting  forms  are 


P'  =  ^>uuu  a        ,    ,          ,  (306) 

T7-  I    *    /T^2       I         T\2   I 


88,000^  [-=^==1  (307) 


Equations  (306)  and  (307)  give  the  least  thrusts  required  along 
the  shafts  of  the  transmission  in  order  to  transmit  the  given  horse 
power. 

CROWN -FRICTION  GEARING 

Crown-friction  gearing  is  used  to  transmit  power  by  means  of 
shafts  that  intersect  and  are  at  right  angles  to  each  other.  A 
simple  form  of  this  type  of  transmission  as  applied  to  the  driving 
of  a  light  motor  car  is  shown  in  Fig.  123.  In  slightly  modified 


268 


CROWN-FRICTION  GEARING 


[CHAP.  XI 


form  this  same  mechanism  has  been  applied  to  machine  tools  for 
varying  feeds.  Within  recent  years  crown  frictions  have  been 
used  successfully  in  automobile,  motor-truck,  and  tractor  trans- 
missions, as  well  as  in  the  driving  of  screw  power  presses  and 
sensitive  drilling  machines. 

The  wheel  c  in  Fig.  123  is  generally  faced  with  compressed 
paper,  vulcanized  fiber,  leather,  or  other  suitable  friction  material, 


FIG.  123. 

and  is  slightly  crowned  in  order  to  decrease  the  slipping  action 
which  takes  place,  due  to  the  varying  speeds  of  the  points  in  con- 
tact. Now  since  wheel  c  is  made  of  a  softer  material  than  that 
used  on  the  disc  6,  it  should  act  as  the  driver  so  that  its  surface 
will  not  be  worn  flat  at  spots  by  the  rotation  of  the  disc  against 
it  under  pressure.  However,  this  is  not  the  usual  method  of 


ART.  201] 


CROWN-FRICTION  GEARING 


269 


mounting  a  crown-friction  transmission.  As  now  installed,  the 
disc  serves  as  the  driving  member  and  in  practically  all  cases  its 
face  is  plain  cast  iron.  In  the  design  just  mentioned,  the  speed 
of  the  wheel  c  may  be  varied  by  simply  moving  c  across  the  face  of 
the  disc,  while  the  direction  of  rotation  of  the  wheel  may  be  re- 
versed by  moving  it  clear  across  the  center  of  the  disc. 

201.  Force  Analysis. — To  determine  the  forces  acting  upon  the 
various  members  of  a  crown-friction  transmission  similar  to  that 
shown  in  Fig.  123,  the  following  method  may  be  used: 

TT 

The  twisting  moment  on  the  driving  shaft  is  63,030  -^»   hence 

the  tangential  forces  acting  upon  the  driven  wheel  for  the  two 
limiting  speeds  are  as  follows: 

A  m        126,060  # 
At  the  minimum  speed,  1 1  =  — ^r — 

*Ul  (308) 


At  the  maximum  speed,  T2  = 


126,060  H 


ND, 


in  which  DI  and  D2  denotes  the  minimum  and  maximum  diam- 
eters of  the  driving  disc,  respectively. 

The  thrusts  that  must  be  applied  to  the  disc  for  the  two  speeds 
are  obtained  by  dividing  the  values  of  TI  and  T2  by  the  coefficient 
of  friction  /*,  giving 


Pi  = 


126,060  H 


126,060  H 


(309) 


The  forces  actually  available  on  the  chain  sprocket  /  for  the  two 
cases  considered  are  as  follows : 


(310) 


in  which  Z)3  denotes  the  diameter  of  the  sprocket  /.  The 
efficiency  77  may  be  taken  as  60  per  cent,  at  the  low  speeds  and  80 
per  cent,  at  the  high  speeds. 

To  determine  the  magnitude  of  the  force  F  required  to  shift 
the  driven  wheel  c,  multiply  the  thrust  P  exerted  by  the  disc  6 
upon  the  wheel  c  by  the  coefficient  of  friction  /x  and  add  to  this  the 


270 


CROWN-FRICTION  GEARING 


[CHAP.  XI 


force  required  to  overcome  the  frictional  resistance  between 
the  wheel  c  and  its  shaft  e.     Whence 

F=P(/i  +  Mi),  (311) 

in  which  MI  denotes  the  coefficient  of  friction  between  the  wheel 
c  and  its  shaft  e. 

202.  Pressures  on  the  Various  Shaft  Bearings.— (a)  Driving 
shaft. — The  various  forces  discussed  in  Art.  201  produce  pres- 
sures upon  the  several  bearings  used  in  the  transmission.  The 
same  type  of  crown  friction  drive  shown  in  Fig.  123  is  represented 


T 


FIG.  124. 

diagrammatically  in  Fig.  124.  The  journal  A  on  the  driving 
shaft  a,  due  to  the  tangential  force  TI  and  the  thrust  P  at  the 
point  of  contact  is  subjected  to  the  following  pressures: 

PD 

1.  A  horizontal  force  due  to  P,  having  a  magnitude  of  -^ — 

This  result  is  obtained  by  taking  moments  about  axis  of  the 
journal  B. 

2.  A  vertical  force,   due  to  TI,  the  magnitude  of  which  is 
obtained    as    in    the    preceding    case.     This    pressure    acts   in 
the  same  direction  as  the  tangential   force  TI  and  its  magni- 
tude is  —Ti. 

m 

By  a  similar  analysis  the  following  forces  acting  upon  the  jour- 
nal B  are  determined: 


ART.  203]  FRICTION  SPINDLE  PRESS  271 

PD 

3.  A  horizontal  force  equal  to  "o"^* 

m 

4.  A  vertical  force  equal  to  — -(m  +  n). 

m  v  ' 

From  the  analysis  of  the  forces  acting  upon  the  shaft  a,  it  is 
evident  that  at  the  point  B  this  shaft  is  subjected  to  a  bending 
stress  in  addition  to  a  torsional  stress.  There  may  also  be  a  com- 
pressive  stress  due  to  the  thrust  P,  but  this  can  be  avoided  by  a 
careful  arrangement  of  the  thrust  bearing  at  that  point.  To  de- 
termine the  size  of  the  shaft,  use  the  principles  discussed  in  the 
chapter  on  shafting. 

(b)  Driven  shaft. — The  driven  shaft  e  is  subjected  to  a  combined 
torsion  and  bending  between  the  wheel  c  and  the  sprocket  /. 
The  wheel  c  is  acted  upon  by  the  two  forces  P  and  TI,  the  former 
producing  pure  bending  of  the  shaft  and  the  latter  combined 
torsion  and  bending.  After  having  calculated  the  load  on  the 
sprocket  /,  the  pressures  upon  the  bearings  C  and  E  may  be 
obtained  by  an  analysis  similar  to  that  used  above. 

203.  Friction  Spindle  Press. — The  so-called  friction  spindle 
press  used  to  a  large  extent  in  Germany  is  an  excellent  application 
of  crown-friction  gearing.  In  this  country,  the  Zeh  and  Hahne- 
mann  Co.  of  Newark,  N.  J.  are  about  the  only  manufacturers  that 
have  introduced  friction  gearing  on  presses  for  forging  and  stamp- 
ing operations.  One  of  their  designs  is  shown  in  Fig.  125.  The 
friction  wheel  d  is  really  a  heavy  flywheel  fastened  rigidly  to  the 
vertical  screw  e.  The  face  of  the  flywheel  is  grooved,  and  into 
this  groove  is  fitted  a  leather  belt  which  serves  as  the  friction 
medium.  The  driving  shaft  a  is  equipped  with  two  plain  cast- 
iron  discs  b  and  cf  which  may  be  brought  into  contact  with  the 
friction  wheel  d  by  moving  the  entire  shaft  endwise.  It  should 
be  understood  that  the  function  of  the  friction  drive  is  merely  to 
accelerate  the  flywheel  d,  and  the  energy  stored  up  during  the 
accelerating  period  does  the  useful  work.  To  accelerate  the  fly- 
wheel, the  driving  shaft  is  moved  endwise  against  the  action  of 
the  spring  /  until  b  is  in  contact  with  d,  thus  causing  the  screw  to 
rotate.  This  rotation  causes  the  screw  and  attached  flywheel  to 
move  downward,  increasing  its  rotative  speed  as  well  as  that  in  a 
downward  direction.  It  should  be  noted  that  the  flywheel  gen- 
erates a  spiral  on  the  face  of  the  disc  b.  At  the  end  of  the  working 
stroke  of  the  screw  a  suitable  tappit,  located  on  the  crosshead  at 
the  lower  end  of  the  screw,  operates  a  linkage  which  disengages 


272 


FRICTION  SPINDLE  PRESS 


[CHAP.  XI 


6  and  d,  thus  permitting  the  spring /to  force  the  disc  c  against  the 
flywheel  d  causing  it  to  return  to  the  top  of  the  stroke. 

This  type  of  press  is  especially  adapted  for  work  in  which  a  hard 
end  blow  is  required.  It  is  not  suitable  for  work  requiring  a 
heavy  pressure  through  a  considerable  part  of  the  stroke,  such  as 
is  required  in  the  manufacture  of  shells,  for  example. 


FIG.  125. 

204.  Curve  Described  by  the  Flywheel. — In  discussing  the 
action  of  the  friction  spindle  press,  it  is  of  interest  to  investigate 
the  nature  of  the  path  or  curve  described  by  the  flywheel  on  the 
face  of  the  friction  disc.  It  is  apparent  that  the  tangential 
velocity  vt  of  the  flywheel  rim  is  proportional  to  the  radius  of  the 
driving  disc;  hence  at  any  point  a  distance  r  from  the  center  of 
the  driving  shaft,  the  magnitude  of  this  velocity  is 


=  cr 


(312) 


The  velocity  va  of  the  screw  in  a  direction  parallel  to  its  center 
line  also  is  proportional  to  the  radius  r;  hence 


va  =  kr 


(313) 


Combining  (312)  and  (313),  it  follows  that  the  ratio  of  v.  to 
vt  is  a  constant,  the  value  of  which  is  readily  determined.  Rep- 
resenting the  diameter  of  the  flywheel  by  D  and  the  lead  of  the 


ART.  205] 


FRICTION  SPINDLE  PRESS 


273 


screw  by  p,  both  being  expressed  in  inches,  the  relation  existing 
between  va  and  vt,  is 


from  which 


v8        p 

-    =   — ~    =   A 

vt        irD 


(314) 


Let  ABC  of  Fig.  126  represent  a  part  of  the  curve  described  by 
the  flywheel  on  the  surface  of  the  driving  disc;  then 

rdd        1 


FIG.  126. 


since  the  velocity  v  makes  a  constant  angle  with  the  radius  vector. 
Hence,  we  get  by  integration 

loge  r  =  KB 

(315) 


or 


r  =  e' 


It  appears  that  the  curve  described  by  the  flywheel  in  moving 
across  the  face  of  the  driving  disc  is  an  equiangular  or  logarithmic 
spiral. 

205.  Pressure  Developed  by  a  Friction  Spindle  Press. — (a) 

Working  stroke. — Beginning  with  the  ram  at  the  top  of  the  down 
stroke,  the  friction  wheel  d  being  at  rest  will  tend  to  assume  the 
same  velocity  as  the  driving  disc  6,  but  due  to  slippage  this  con- 
dition will  not  prevail  until  the  screw  and  flywheel  have  moved 
downward  a  certain  distance.  During  the  next  period  the  wheel 
d  is  accelerated  with  practically  no  slippage,  and  when  the  tool 
strikes  the  work,  the  friction  wheel,  the  screw  and  ram  have  ac- 


274  FRICTION  SPINDLE  PRESS  [CHAP.  XI 

cumulated  a  certain  amount  of  energy  which  is  given  out  in  per- 
forming useful  work  during  the  remainder  of  the  stroke.  It 
should  be  noted  that  the  driving  disc  is  thrown  out  of  contact 
with  d  about  the  same  instant  that  the  tool  strikes  the  work; 
hence  the  driving  force  is  not  considered  as  doing  any  useful 
work,  but  is  used  merely  to  accelerate  the  moving  system. 
It  is  evident,  therefore,  that  the  pressure  developed  during  the 
latter  part  of  the  stroke  depends  upon  the  energy  stored  up  by 
the  flywheel,  screw,  and  ram,  and  the  distance  through  which 
the  ram  moves  in  doing  the  work. 

Assuming  that  the  flywheel  d  is  r%  inches  from  the  center  of 
the  driving  shaft  when  the  tool  strikes  the  work,  the  kinetic 
energy  in  the  flywheel  and  screw  due  to  the  tangential  velocity 
Vt  is  given  by  the  following  expression  : 


0  „ 
20 

in  which  W\  is  the  equivalent  weight  of  the  wheel  and  screw  re- 
duced to  the  outside  radius  of  the  rim  having  a  velocity  of 


Vt  =: 


360 


in  which  N  denotes  the  revolutions  per  minute  made  by  the 
driving  shaft. 

Denoting  the  actual  weight  of  the  wheel,  screw  and  ram  by 
the  symbol  W%,  we  find  that  the  energy  stored  up  in  these  parts 
due  to  the  velocity  va  is 


in  which  va,  according  to  (314),  is 

pvt  =    przN 
irD       360  D 

Now  in  coming  to  rest  the  moving  mass  Wz  also  does  external 
work,  the  magnitude  of  which  is 


in  which  rs  denotes  the  distance  between  the  center  line  of  the 
driving  shaft  and  the  flywheel  at  the  end  of  the  downstroke. 


ART.  206]        EFFICIENCY  OF  CROWN-FRICTION  GEARING        275 

Summing  up,  we  find  that  the  theoretical  amount  of  work  that 
can  be  done  is 

E  =  Ei  +  E2  +  E3  (316) 

and  multiplying  this  by  the  efficiency  rj,  the  external  or  useful 
work  that  can  be  done  is  rjE. 

The  average  pressure  Q  upon  the  tool  multiplied  by  the  dis- 
tance — TO —  through  which  this  force  acts  must  be  equivalent  to 
the  work  done  by  the  moving  system;  hence 

(317) 


(b)  Return  stroke. — On  the  return  stroke,  the  driving  disc  c 
is  brought  into  contact  with  the  wheel  d,  and  since  the  latter  is 
at  rest  for  a  short  interval  of  time,  we  have  the  same  conditions 
to  contend  with  that  prevailed  at  the  beginning  of  the  working 
stroke,  namely,  that  the  flywheel  will  slip  until  it  attains  the 
same  speed  as  the  driving  disc.  After  the  flywheel  has  attained 
the  speed  as  the  driving  disc,  this  condition  will  continue  until 
the  disc  c  is  released  and  the  disc  6  is  again  applied. 

206  Double-crown  Frictions. — An  interesting  variable-speed 
friction  drive  used  on  the  Albany  sensitive  drill  press  is  shown  in 
Fig.  127.  It  consists  of  two  crown  friction  wheels,  one  of  which 
is  mounted  on  the  drive  shaft  a,  and  the  other  on  the  spindle  k 
of  the  drill  press.  A  hemisphere  c,  made  of  cast  iron  and  bushed 
with  bronze,  is  mounted  on  a  shaft  d,  which  is  pivoted  on  the 
adjustable  spindle  e.  By  means  of  the  handle  g,  the  shaft  d  and 
the  hemisphere  c  may  be  moved  in  a  vertical  plane.  The  speed 
of  rotation  of  the  hemisphere,  and  the  speed  of  the  driven  wheel 
h  are  thus  varied.  The  contact  pressure  between  the  hemisphere 
and  the  friction  wheels  may  be  increased  or  decreased  by  means 
of  the  adjusting  nut  /.  Ball  bearings  are  used  in  all  of  the 
important  bearings  on  the  machine,  as  shown  in  Fig.  127. 

207.  Efficiency  of  Crown-friction  Gearing. — A  study  of  the 
action  of  crown-friction  gearing  shows  clearly  that  the  points  on 
the  disc  6  in  contact  with  the  inner  and  outer  edges  of  the  driven 
wheel  c  will  travel  unequal  distances  per  revolution  of  the  disc 
(see  Fig.  124).  From  this  it  follows  that  there  is  slippage  be- 


276 


EFFICIENCY  OF  CROWN-FRICTION  GEARING    [CHAP.  XI 


tween  the  wheel  and  the  disc  at  the  line  of  contact.  Denoting 
by  /  the  width  of  the  face  of  the  wheel  c,  then  the  difference 
between  the  distances  traveled  per  revolution  of  the  disc  by  the 
extreme  points  in  contact  is  2irf. 


FIG.  127. 


To  determine  the  work  lost  per  revolution  due  to  slippage 
multiply  the  average  slip  irf  by  the  tangential  resistance  between 
the  wheel  and  the  disc;  thus 

W8  =  wfP  (318) 


The  output  per  revolution  of  the  disc  is  juTrPD;  hence  the  total 
work  put  in,  exclusive  of  that  required  to  overcome  the  frictional 
resistances  of  the  various  bearings,  is  given  by  the  expression 


W  = 


(319) 


ART.  208]  MOUNTING  FRICTION  GEARING  277 

Since   the   efficiency   of  any  machine  is  equal  to  the  output 
divided  by  the  input,  we  obtain  in  this  case 

V  =  o%rf  (320) 

According  to  (320),  the  efficiency  of  crown-friction  gearing  is 
independent  of  the  diameter  of  the  driven  wheel.  Furthermore, 
the  efficiency  increases  as  the  face  of  the  crown  wheel  is  decreased, 
and  as  the  line  of  contact  between  the  disc  and  the  wheel  is  moved 
farther  from  the  center  of  the  disc. 

.  MOUNTING  FRICTION  GEARING 

In  general,  friction  gearing  must  be  mounted  in  such  a  manner 
that  the  pressure  required  between  the  surfaces  in  contact  in 
order  to  transmit  the  desired  horse  power  can  readily  be  pro- 
duced. This  result  is  obtained  by  equipping  one  of  the  shafts 
with  a  special  bearing  or  set  of  bearings. 

208.  Thrust  Bearings  for  Friction  Gearing. — (a)  Bearings  for 
spur  and  grooved  frictions. — In  the  case  of  spur  and  grooved  fric- 
tion gearing,  the  pinion  shaft  is  mounted  on  eccentric  bearings,  the 
constructive  details  of  which  are  shown  clearly  in  Fig.  128.  The 
gears  themselves  should  be  located  close  to  the  bearings  in  order 
to  insure  rigidity,  thus  obviating  undue  wear  on  the  gears  as  well 
as  on  the  bearings. 

(b)  Thrust  bearings  for  bevel  frictions. — For  engaging  a  pair  of 
bevel  gears,  and  taking  up  any  wear  that  may  occur,  two 
types  of  bearings  are  in  common  use.  The  first  type,  which 
may  be  called  a  quick-acting  end-thrust  bearing,  is  shown  in 
Fig.  129.  It  is  used  in  connection  with  bevel  frictions  requiring 
frequent  throwing  in  and  out  of  engagement.  The  inner  sleeve 
a  forming  the  bearing  for  the  shaft  has  a  helical  slot  into  which  the 
turned  end  of  the  adjusting  screw  b  is  fitted.  It  is  evident  that 
rotating  the  sleeve  a  in  the  proper  direction  will  cause  the  sleeve 
to  advance  in  an  axial  direction,  thus  engaging  the  gears. 

The  second  type  of  end  thrust  bearing  works  on  the  same  general 
principle.  The  inner  sleeve,  instead  of  being  fitted  with  a  heli- 
cal slot,  is  threaded  as  shown  in  Fig.  130.  This  design  is  well 
adapted  to  installations  in  which  the  friction  gears  are  not  en- 
gaged or  disengaged  very  frequently. 


278 


MOUNTING  FRICTION  GEARING 


[CHAP.  XI 


(c)  Thrust  bearings  for  crown  frictions. — For  engaging  crown 
frictions,  the  same  type  of  bearings  as  those  shown  in  Figs.  129 


FIG.  128. 


and  130  may  be  used.     Occasionally,  spring  thrust  bearings  are 
used  in  place  of  those  just  mentioned. 


FIG.  129. 


209.  Starting  Loads. — As  stated  in  Art.  194,  the  coefficient  of 
friction  is  a  maximum  when  the  slip  between  the  friction  gears 


FIG.  130. 


lies  between  2  and  6  per  cent.  Experiments  have  also  shown  that 
the  coefficient  of  friction  decreases  gradually  as  the  slip  increases ; 
hence  when  a  friction  transmission  is  started  under  load,  the 


ART.  209]  REFERENCES  279 

pressure  that  must  be  applied  to  the  surfaces  in  contact  is  from 
two  to  three  times  as  great  as  that  required  for  normal  operation. 
This  is  due  to  the  decrease  in  the  coefficient  of  friction  caused  by 
the  excessive  slippage  during  the  period  of  starting.  From  this 
discussion  it  follows  that  the  bearings  described  in  the  preceding 
paragraphs  must  be  designed  for  the  starting  conditions.  After 
the  transmission  is  once  started  the  thrusts  on  the  gears  may  be 
reduced  considerably,  thus  eliminating  excessive  wear  and  lost 
work. 

References 

Machine  Design,  Construction  and  Drawing,  by  SPOONER. 

Paper  Friction  Wheels,  Trans.  A.  S.  M.  E.,  vol.  18,  p.  102. 

Friction  Driven  Forty-four  Foot  Pit  Lathe,  Trans.  A.  S.  M.  E.,  vol.  24, 
p.  243. 

Power  Transmission  by  Friction  Driving,  Trans.  A.  S.  M.  E.,  vol.  29,  p. 
1093. 

Efficiency  of  Friction  Transmission,  The  Horseless  Age,  July  6,  1910. 

Friction  Transmission,  The  Rockwood  Mfg.  Co. 


CHAPTER  XII 
SPUR  GEARING 

Friction  gearing,  as  has  been  stated,  is  not  suitable  for  the 
transmission  of  large  amounts  of  power,  nor  where  it  is  desir- 
able that  the  velocity  ratio  between  the  driving  and  driven  mem- 
bers be  absolutely  positive.  For  such  a  transmission  it  becomes 
necessary  to  provide  the  surfaces  in  contact  with  grooves  and 
projections,  thus  providing  a  positive  means  of  rotation.  The 
original  surfaces  of  the  frictions  then  become  the  so-called  pitch 
surfaces  of  the  toothed  gears,  and  the  projections  together  with 
the  grooves  form  the  teeth.  These  teeth  must  be  of  such  a  form 
as  to  satisfy  the  following  conditions: 

(a)  The  teeth  must  be  capable  of  transmitting  a  uniform  ve- 
locity ratio.     The  condition  is  met  if  the  common  normal  at 
the  point  of  contact  of  the  tooth  profiles  passes  through  the 
pitch  point,  i.e'.,  the  point  of  tangency  of  the  two  pitch  lines. 

(b)  The  relative  motion  of  one  tooth  upon  the  other  should  be 
as  much  a  rolling  motion  as  possible  on  account  of  the  greater 
friction  and  wear  attendant  to  sliding.     With  toothed  gearing, 
however,  it  is  impossible  to  have  pure  rolling  contact  and  still 
maintain  a  constant  velocity  ratio. 

(c)  The  tooth  should  conform  as  nearly  as  possible  to  a  canti- 
lever beam  of  uniform  strength,  and  should  be  symmetrical  on 
both  sides  so  that  the  gear  may  run  in  either  direction. 

(d)  The  arc  of  action  should  be  rather  long  so  that  more  than 
one  pair  of  teeth  may  be  in  mesh  at  the  same  time. 

210.  Definitions.^Before  taking  up  the  discussion  of  the 
various  types  of  tooth  curves,  it  is  well  to  familiarize  ourselves 
with  the  meaning  of  different  terms  and  expressions  used  in  con- 
nection with  gearing  of  all  kinds. 

(a)  By  the  term  circular  pitch  is  meant  the  distance  from  one 
tooth  to  a  corresponding  point  on  the  next  tooth,  measured  on  the 
pitch  circle.  The  circular  pitch  is  equal  to  the  circumference  of 
the  pitch  circle  divided  by  the  number  of  teeth  in  the  gear. 

(6)  The  diametral  pitch  is  equal  to  the  number  of  teeth  in  the 

280 


ART.  210] 


DEFINITIONS 


281 


gear  divided  by  the  pitch  diameter.     It  is  not  a  dimension  on  the 
gear,  but  is  simply  a  convenient  ratio. 

(c)  The  term  chordal  pitch  may  be  defined  as  the  distance  from 
one  tooth  to  a  corresponding  point  on  the  next  measured  on  a 
chord  of  the  pitch  circle  instead  of  on  the  circumference.     This 
pitch  is  used  only  in  making  the  drawing  or  by  the  pattern  maker 
if  the  teeth  are  to  be  formed  on  a  wood  pattern. 

(d)  The  thickness  of  the  tooth  is  the  thickness  measured  on  the 
pitch  line,  as  illustrated  in  Fig.  131. 

(e)  By  the  tooth  space  is  meant  the  width  of  the  space  on  the 
pitch  line. 

(/)  The  term  backlash  means  the  difference  between  the  tooth 
space  and  the  thickness  of  the  tooth. 


ClC  Pitch    • 

.  Chorda!  Pit* -~\ 
Face        \ 


FIG.  131. 

(g)  By  the  term  addendum  is  meant  the  distance  from  the  pitch 
circle  to  the  ends  of  the  teeth,  as  dimension  a  in  Fig.  131. 

(h)  The  distance  6  between  the  pitch  circle  and  the  bottom  of 
the  tooth  space  is  called  the  dedendum. 

(i)  The  clearance  is  the  difference  between  the  addendum  and 
the  dedendum,  or  in  other  words,  the  amount  of  space  between 
the  root  of  a  tooth  and  the  point  of  the  tooth  that  meshes  with  it. 

(j)  As  shown  in  Fig.  131,  the  face  of  the  tooth  is  that  part  of 
the  tooth  profile  which  lies  between  the  pitch  circle  and  the 
end  of  the  tooth. 

(k)  The  flank  of  the  tooth  is  that  part  of  the  tooth  profile  which 
lies  between  the  pitch  circle  and  the  root  of  the  tooth,  as 
represented  in  Fig.  131. 

(I)  The  line  of  centers  is  the  line  passing  directly  through  both 
centers  of  a  pair  of  mating  gears. 


282 


TOOTH  CURVES 


[CHAP.  XII 


(m)  The  pitch  circles  of  a  pair  of  gears  are  imaginary  circles, 
the  diameters  of  which  are  the  same  as  the  diameters  of  a  pair  of 
friction  gears  that  would  replace  the  spur  gears. 

(n)  The  base  circle  is  an  imaginary  circle  used  in  involute  gear- 
ing to  generate  the  involutes  which  form  the  tooth  profiles.  It 
is  drawn  tangent  to  the  line  representing  the  tooth  thrust,  as 
shown  in  Fig.  131. 

(0)  The  describing  circle  is  an  imaginary  circle  used  in  cycloidal 
gearing  to  generate  the  epicycloidal  and  hypocycloidal  curves 
which  form  the  tooth  profiles.     There  are  two  describing  circles, 
one  inside  and  one  outside  of  the  pitch  circle,  and  they  are  usu- 
ally of  the  same  size. 

(p)  By  the  angle  of  obliquity  of  action  is  meant  the  inclination 
of  the  line  of  action  of  the  pressure  between  a  pair  of  mating  teeth 
to  a  line  drawn  tangent  to  the  pitch  circle  at  the  pitch  point,  as 

represented  in  Fig.  131  by 
the  angle  a  or  the  angle 
DCF  in  Fig.  132. 

(q)  The  arc  of  approach  is 
the  arc  measured  on  the 
pitch  circle  of  a  gear  from 
the  position  of  the  tooth  at 
the  beginning  of  contact  to 
the  central  position,  that  is, 
the  arc  HC  in  Fig.  132. 

(r)  The  arc  of  recess  is  the 
arc  measured  on  the  pitch  circle  from  the  central  position  of 
the  tooth  to  its  position  where  contact  ends,  that  is,  the  arc  CI 
in  Fig.  132. 

(s)  The  arc  of  action  is  the  sum  of  the  arcs  of  approach  and 
recess. 

(1)  By  the  term  velocity  ratio  is  always  meant  the  ratio  of  the 
number  of  revolutions  of  the  driver  to  the  number  of  revolutions 
of  the  driven  gear. 

211.  Tooth  Curves. — There  are  many  different  types  of  curves 
that  would  serve  as  profiles  for  teeth  and  satisfy  the  condition 
of  constant  velocity  ratio,  with  sufficient  accuracy  for  all 
practical  purposes;  but  there  are  in  actual  use  only  two, 
namely,  the  involute  and  the  cycloidal.  As  regards  strength  and 
efficiency  the  two  forms  are  practically  on  a  par.  However,  the 


FIG.  132. 


ART.  212]  METHODS  OF  MANUFACTURE  283 

involute  tooth  has  one  decided  advantage  over  the  cycloidal, 
namely,  that  the  distance  between  centers  may  be  slightly  greater 
or  less  than  the  theoretical  distance  without  affecting  the  velocity 
ratio.  The  cycloidal  tooth,  also,  has  one  important  advantage 
over  the  involute,  namely,  that  a  convex  surface  is  always  in 
contact  with  one  that  is  concave.  Although  the  contact  is 
theoretically  a  line,  practically  it  is  not ;  consequently,  the  wear  is 
not  so  rapid  as  with  involute  teeth  where  the  surfaces  are  either 
convex  or  straight. 

212.  Methods  of  Manufacture. — Gear  teeth  are  formed  in 
practice  by  two  distinct  processes,  moulding  and  machine  cutting. 
Formerly  all  gears  were  cast  and  the  moulds  were  formed  from 
complete  patterns  of  the  gears.  Of  late  years,  however,  gear 
moulding  machines  are  used  to  a  considerable  extent,  and  the 
results  obtained  are  superior  to  the  pattern-moulded  gear. 
Even  with  machine  moulding,  however,  the  teeth  are  somewhat 
rough  and  warped  out  of  shape,  so  that  the  gears  always  run  with 
considerable  friction  and  are  not  suited  to  high  speeds.  At  the 
present  time  gears  of  ordinary  size  are  almost  always  cut,  except 
those  used  in  the  cheaper  class  of  machinery.  The  method  which 
is  commonly  used  is  to  cut  the  teeth  with  a  milling  cutter  that 
has  been  formed  to  the  exact  shape  required.  There  are  also  two 
styles  of  gear  planers,  one  of  which  generates  mathematically 
correct  profiles  by  virtue  of  the  motion  given  to  the  cutter  and 
the  gear  blank,  and  the  other  forms  the  outlines  by  following 
a  previously  shaped  templet.  Another  method  of  producing 
machine  cut  teeth  is  by  the  stamping  process  now  used  extensively 
in  the  manufacture  of  gears  for  clocks,  slot  machines,  etc. 

A  method  of  generating  spur  and  helical  gear  teeth  by  means 
of  a  hob  is  now  recognized  and  accepted  as  the  best  way  of  pro- 
ducing accurate  teeth.  In  this  generating  process  a  hob, 
threaded  to  the  required  pitch,  is  rotated  in  conjunction  with  the 
gear  blank  at  a  ratio  dependent  upon  the  number  of  teeth  to  be 
cut.  The  cross-section  of  the  thread  is  a  rack  that  will  mesh 
correctly  with  the  gear  to  be  cut.  One  important  advantage  of 
this  process  is  that  only  one  hob  is  required  for  cutting  all  num- 
bers of  teeth  of  one  given  pitch.  Another  advantage  of  the  nob- 
bing system  is  that  gears  can  be  produced  more  cheaply  than  by 
any  other  system. 


284  INVOLUTE  SYSTEM  [CHAP.  XII 

SYSTEMS  OF  GEAR  TEETH 

213.  Involute  System. — In  the  involute  system  of  gearing  the 
outline  of  the  tooth  is  an  involute  of  a  circle,  called  the  base 
circle.  However,  when  the  tooth  extends  below  the  base  circle 
that  portion  of  the  profile  is  made  radial.  The  simplest  concep- 
tion of  an  involute  is  as  follows:  if  a  cord,  which  has  previously 
been  wound  around  any  given  plane  curve  and  has  a  pencil  at- 
tached to  its  free  end,  is  unwound,  keeping  the  cord  perfectly 
tight,  the  pencil  will  trace  the  involute  of  the  given  curve.  The 
base  circle  may  easily  be  obtained  by  drawing  through  the  pitch 
point  a  line  making  an  angle  with  the  tangent  to  the  pitch  circle 
at  this  point,  equal  to  the  angle  of  obliquity  of  action;  then  the 
circle  drawn  tangent  to  this  line  will  be  the  required  base  circle. 

In  order  to  manufacture  gears  economically,  it  is  essential  that 
any  gear  of  a  given  pitch  should  work  correctly  with  any  other 
gear  of  the  same  pitch,  thus  making  an  interchangeable  set.  To 
accomplish  this  end,  standard  proportions  have  been  adopted  for 
the  teeth. 

(a)  Angle  of  obliquity. — The  angle  of  obliquity  of  action  which 
is  generally  accepted  as  the  standard  for  cast  teeth  is  15  degrees, 
although  in  cases  of  special  design  this  angle  is  often  made  greater. 
When  the  angle  of  obliquity  is  increased,  the  component  of  the 
pressure  tending  to  force  the  gears  apart  and  producing  friction 
in  the  bearings  is  increased;  but  on  the  other  hand,  the  profile  of 
the  tooth  becomes  wider  at  the  base  and  consequently  the 
strength  is  correspondingly  greater.  Such  gears,  having  large 
angles  of  obliquity,  are  used  where  the  conditions  are  unusual  and 
where  the  standard  tooth  form  is  not  suitable.  In  England, 
teeth  of  greater  obliquity  of  action  and  less  depth  than  the  stand- 
ard are  quite  common,  and  at  present  there  is  a  tendency  in 
that  direction  in  America.  For  cut  teeth  now  used  in  motor- 
car construction  as  well  as  in  machine  tools,  the  manufacturers 
have  adopted  what  is  called  the  stub  tooth,  having  an  angle  of 
obliquity  of  20  degrees.  The  proportions  of  the  teeth  as  used 
for  this  service  are  given  in  Art.  223.  In  designing  teeth  of  the 
stub-tooth  form,  care  must  be  taken  to  make  the  arc  of  action  at 
least  as  great  as  the  circular  pitch;  otherwise  the  teeth  would 
not  be  continuously  in  mesh  and  would  probably  come  to- 
gether in  such  a  way  as  to  lock  and  prevent  further  rota- 
tion. The  standard  angle  of  obliquity  of  action,  adopted  by 


ART.  214]          LAYING  OUT  THE  INVOLUTE  TOOTH  285 

manufacturers  of  gear  cutters  and  used  almost  exclusively  before 
the  advent  of  the  stub  teeth,  is  slightly  at  variance  with  the 
usual  standard  for  cast  teeth,  being  14°  28'  40",  the  sine  of  which 
is  0.25. 

(b)  Smallest  number  of  teeth. — The  smallest  involute  gear  of 
standard  proportions  that  will  mesh  correctly  with  a  rack  of  the 
same  pitch  contains  30  teeth;  however,  this  difficulty  is  met  by 
slightly  correcting  the  points  of  all  the  teeth  in  the  set,  so  that 
a  gear  of  12  teeth  may  mesh  with  any  of  the  other  gears  of  the 
same  pitch.  The  profiles  of  the  teeth  may  be  drawn  accurately 
by  means  of  circular  arcs  having  their  centers  on  the  base  circle 
B,  as  shown  in  Fig.  133.  The  value  of  these  radii  for  a!5-degree 
involute  have  been  carefully  worked  out  by  Mr.  G.  B.  Grant  of 
the  Grant  Gear  Works  and  are  given  in  Table  65. 

214.  Laying  Out  the  Involute  Tooth. — To  apply  the  tabular 
values  given  in  Table  65,  draw  the  pitch,  addendum  and  clearance 
circles  in  the  usual  way,  and  space  off  the  pitch  of  the  teeth  on  the 
pitch  circle.  The  base  circle  is  constructed  next.  This  may  be 
done  as  described  in  a  preceding 
article  or  by  making  the  distance  a 
in  Fig.  133  equal  to  one-sixtieth  of 
the  pitch  diameter.  With  the  base 
line  B  as  a  circle  of  centers,  draw 
that  part  of  the  tooth  profile  above 
the  pitch  line  A,  generally  called  FIG  133 

the  face  of  the  tooth,  by  using  the 

face  radius  b  given  in  Table  65.  Next  draw  in  that  part  of  the 
tooth  profile  between  the  pitch  line  A  and  the  base  circle,  using 
the  flank  radius  c  given  in  Table  65.  To  finish  the  tooth,  that 
part  lying  between  the  base  circle  and  the  fillet  at  the  root  of  the 
tooth  is  made  a  radial  line,  as  shown  in  Fig.  133.  It  should  be 
noticed  that  the  values  of  b  and  c  given  in  Table  65  are  for  1 
diametral  pitch  or  1  inch  circular  pitch.  For  any  other  pitch 
divide  or  multiply  the  tabular  values  by  the  given  pitch  as 
directed  in  the  table. 

It  will  be  noted  that  the  tabular  values  in  this  table  are  for 
15-degree  involutes  and  therefore  do  not  apply  to  the  standard 
form  of  cut  teeth.  The  forms  given,  however,  may  be  used  on 
the  drawing,  because  in  cutting  a  gear  the  workman  needs  to 
know  only  the  number  of  teeth  in  the  gear,  and  either  the  number 
of  the  cutter  or  the  pitch  of  the  hob.  All  that  is  required  on  a 


286 


GRANT'S  TABLE  FOR  INVOLUTE  TEETH       [CHAP.  XII 


drawing  is  an  approximate  representation  of  the  tooth  profile. 
The  table  also  gives  values  down  to  a  10-tooth  gear,  while  the 
standard  cut  gear  sets  run  down  to  12  teeth  only.  This  is  theo- 
retically the  smallest  standard  involute  gear  that  will  have  an 
arc  of  action  equal  to  the  circular  pitch;  however,  in  the  10-  and 
11-tooth  gears  the  error  is  so  slight  that  it  is  practically  un- 
noticeable. 

TABLE  65. — RADII  FOR  IS-DEGREE  INVOLUTE  TEETH 
ACCORDING  TO  G.  B.  GRANT 


No.  of 
teeth 

Divide  by  the 
diametral 
pitch 

Multiply  by 
the  circular 
pitch 

No.  of 
teeth 

Divide  by  the 
diametral 
pitch 

Multiply  by 
the  circular 
pitch 

Rad.  b 

Rad.  c 

Rad.  b 

Rad.  c 

Rad.  b 

Rad.  c 

Rad.  6 

Rad.  c 

10 

2.28 

0.69 

0.73 

0.22 

28 

3.92 

2.59 

.25 

0.82 

11 

2.40 

0.83 

0.76 

0.27 

29 

3.99 

2.67 

.27 

0.85 

12 

2.51 

0.96 

0.80 

0.31 

30 

4.06 

2.76 

.29 

0.88 

13 

2.62 

.09 

0.83 

0.34 

31 

4.13 

2.85 

.31 

0.91 

14 

2.72 

.22 

0.87 

0.39 

32 

4.20 

2.93 

.34 

0.93 

15 

2.82 

.34 

0.90 

0.43 

33 

4.27 

3.01 

.36 

0.96 

16 

2.92 

.46 

0.93 

0.47 

34 

4.33 

3.09 

.38 

0.99 

17 

3.00 

.58 

0.96 

0.50 

35 

4.39 

3.16 

.39 

1.01 

18 

3.12 

.69 

0.99 

0.54 

36 

4.45 

3.23 

.41 

1.03 

19 

3.22 

.79 

.03 

0.57 

37-40 

4.20 

1.34 

20 

3.32 

.89 

.06 

0.61 

41-45 

4.63 

1.48 

21 

3.41 

.98 

.09 

0.63 

46-51 

5.06 

1.61 

22 

3.49 

2.06 

.11 

0.66 

52-60 

5.74 

1.83 

23 

3.57 

2.15 

.13 

0.69 

61-70 

6.52 

2.07 

24 

3.64 

2.24 

.16 

0.71 

71-90 

7.72 

2.46 

25 

3.71 

2.33 

1.18 

0.74 

91-120 

9.78 

3.11 

26 

3.78 

2.42 

1.20 

0.77 

121-180 

13.38 

4.26 

27 

3.85 

2.50 

1.23 

0.80 

181-360 

21.62 

6.88 

(b)  Laying  out  the  rack  tooth. — It  was  found  necessary  to  devise 
a  separate  means  of  drafting  the  rack.  The  tooth  is  drawn  in  the 
usual  manner,  the  sides  from  the  root  line  to  a  point  midway  be- 
tween the  pitch  and  the  addendum  lines  making  angles  of  75 
degrees  with  the  pitch  line.  The  outer  half  of  the  face  is  formed 
by  a  circular  arc  with  its  center  on  the  pitch  line  and  its  radius 
equal  to  2.10  inches  divided  by  the  diametral  pitch,  or  0.67 
multiplied  by  the  circular  pitch.  The  radius  of  the  fillet  at  the 
root  of  the  tooth  is  taken  as  one-seventh  of  the  widest  part  of 
the  tooth  space. 


ART.  215]  INVOLUTE  CUTTERS  287 

215.  Standard   Involute    Cutters. — Brown   and    Sharpe,    the 
leading  manufacturers  of  formed  gear  cutters  in  this  country, 
furnish  involute  cutters  in  sets  of  eight  for  each  pitch,  as  shown  in 
Table  66. 

TABLE  66. — BROWN  AND  SHARPE  STANDARD  INVOLUTE  CUTTERS 

Cutter  No.  1  will  cut  gears  from  135  teeth  to  a  rack. 

Cutter  No.  2  will  cut  gears  from  55  teeth  to  134  teeth. 

Cutter  No.  3  will  cut  gears  from  35  teeth  to    54  teeth. 

Cutter  No.  4  will  cut  gears  from  26  teeth  to    34  teeth. 

Cutter  No.  5  will  cut  gears  from  21  teeth  to    25  teeth. 

Cutter  No.  6  will  cut  gears  from  17  teeth  to    20  teeth. 

Cutter  No.  7  will  cut  gears  from  14  teeth  to    16  teeth. 

Cutter  No.  8  will  cut  gears  from  12  teeth  to    13  teeth. 

When  more  accurate  tooth  forms  are  desired  they  also  furnish 
to  order  cutters  of  the  half  sizes,  making  a  set  of  fifteen  instead  of 
eight  cutters. 

All  of  the  above  cutters  are  commonly  based  on  the  diametral 
pitch  and  are  made  in  the  following  sizes: 

From    1  to    4  diametral  pitch,  the  pitch  advances  by  quarters. 

From    4  to    6  diametral  pitch,  the  pitch  advances  by  halves. 

From  6  to  16  diametral  pitch,  the  pitch  advances  by  whole 
numbers. 

From  16  to  32  diametral  pitch,  the  pitch  advances  by  even 
numbers. 

Then  36,  38,  40,  44,  48,  50,  56,  60,  64,  70,  80,  and  120  diametral 
pitch. 

At  a  slightly  greater  cost,  cutters  based  on  circular  pitch  may  be 
obtained,  and  the  sizes  vary  as  follows: 

From  1  to  IJ^-inch  circular  pitch,  the  pitch  advances  by  J^- 
inch  increments. 

From  1J-2  to  3-inch  circular  pitch,  the  pitch  advances  by  J£- 
inch  increments. 

216.  Action  of  Involute  Teeth. — Fig.  134  illustrates  the  action 
of  a  pair  of  involute  teeth.     Let  the  circles  a  and  6  represent  the 
base  circles  of  a  pair  of  involute  gears,  the  pitch  circles  of  which 
would  be  the  circles  described  about  the  centers  A  and  B  with 
radii  of  AC  and  BC  respectively.     Imagine  a  cord  attached  to  a 
extending  around  the  circumference  to  a  point  D,  from   there 
directly  across  to  E  and  around  the  circumference  of  b.    Let  the 
central  point  of  the  cord  be  permanently  marked  in  some  manner 


288 


CYCLOIDAL  SYSTEM 


[CHAP.  XII 


and  be  denoted  by  C.  Now  rotate  a  in  the  direction  of  the  arrow 
and  trace  the  path  of  the  point  C  on  the  surface  of  a  extended,  on 
the  surface  of  b  extended,  and  also  its  actual  path  in  space.  It  is 
evident  that  these  three  curves  will  be  CG,  CH,  and  CJ,  and  that 
CG  and  CH  will  be  parts  of  the  involutes  of  the  two  base  circles 
a  and  b.  Now  reverse  the  rotation  of  B  and  rewind  the  string  on 
6  until  C  reaches  the  point  K.  During  this  motion  it  will  com- 
plete the  tooth  forms  CF  and  FI.  Bearing  in  mind  that  C  is 
always  the  point  of  contact  of  the  teeth,  its  path  is  evidently  JK 
and  coincides  exactly  with  the  line  of  pressure  between  the 
teeth,  since  the  line  CD  is  always  normal  to  the  involute  curve  it 


FIG.  134. 

is  generating.  If  the  centers  A  and  B  should  be  misplaced 
slightly  on  account  of  wear  in  the  bearings  or  journals,  a  uniform 
velocity  ratio  would  still  be  transmitted  because  the  normals 
would  still  pass  through  the  point  C.  The  shifting  of  the  centers 
will  result  in  a  change  of  the  obliquity  of  the  pressure  on  the  teeth, 
and  the  length  of  the  arc  of  contact.  9  The  outlines  of  the  teeth 
would  not  be  changed  in  the  least. 

217.  Cycloidal  System. — The  cycloidal  system,  although  the 
oldest,  is  not  so  popular  as  the  involute  system  and  seems  to  be 
gradually  going  out  of  use.  Mr.  Grant  in  his  "  Treatise  on  Gear 
Wheels"  says:  " There  is  no  more  need  for  two  different  kinds  of 
tooth  curves  for  gears  of  the  same  pitch  than  there  is  need  for 
different  kinds  of  threads  for  standard  screws,  or  of  two  different 
kinds  of  coins  of  the  same  value,  and  the  cycloidal  tooth  would 
never  be  missed  if  it  were  dropped  altogether.  But  it  was  the 
first  in  the  field,  is  simple  in  theory,  is  easily  drawn,  has  the  recom- 
mendation of  many  well-meaning  teachers,  and  holds  its  position 


ART.  218]  CYLCOIDAL  TOOTH  FORM  289 

by  means  of  human  inertia;  or  the  natural  reluctance  of  the  aver- 
age human  mind  to  adopt  a  change,  particularly  a  change  for  the 
better. "  This  view  is  probably  a  little  biased,  but  nevertheless 
there  is  a  great  deal  of  sound  truth  in  it.  The  proportion  of 
machine  cut  cycloidal  teeth  to  machine  cut  involute  teeth  is  very 
small,  but  in  some  classes  of  work,  and  especially  when  the  loads 
are  heavy,  the  cycloidal  forms  are  still  used  extensively. 

218.  Form  of  the  Cycloidal  Tooth.— The  outline  of  a  cycloidal 
tooth  is  made  up  of  two  curves.  The  faces  of  the  teeth  are  epi- 
cycloids and  the  flanks  are  hypocycloids,  with  two  exceptions, 
namely,  internal  gearing  and  racks.  In  the  former  case,  the  faces 
are  hypocycloids  and  the  flanks  are  epicycloids,  while  in  the  latter 
both  curves  are  plain  cycloids.  When  a  circle  rolls  on  a  fixed 
straight  line,  the  path  generated  by  an  assumed  point  of  the  circle 
is  a  cycloid;  should  the  circle  roll  on  the  outside  of  another  circle, 
the  path  of  this  point  would  be  an  epicycloid,  and  should  it  roll 
on  the  inside  of  another  circle,  it  would  be  a  hypocycloid. 

These  rolling  circles  are  generally  spoken  of  as  describing  cir- 
cles, and  their  size  determines  the  form  of  the  tooth,  the  arc  of 
contact,  and  the  angle  of  obliquity  of  action.  The  angle  of  ob- 
liquity in  the  cycloidal  system  is  constantly  changing;  but  its 
average  value,  when  the  proportions  of  the  teeth  are  standard,  is 
about  15  degrees,  the  same  as  in  involute  gearing.  The  circle 
upon  which  the  describing  circles  are  rolled  is  the  pitch  circle. 
When  the  diameter  of  the  rolling  Sircle  is  equal  to  the  radius  of 
the  pitch  circle,  the  flanks  of  the  teeth  are  undercut.  In  addition 
to  the  objection  that  undercut  teeth  are  weak,  the  amount  of 
undercut  must  be  very  slight  if  the  teeth  are  to  be  cut  with  a 
rotating  cutter. 

The  same  describing  circle  must  always  be  used  for  those  parts 
of  the  teeth  which  work  together,  i.e.,  the  faces  of  a  tooth  on  the 
one  gear  must  be  formed  by  the  same  describing  circle  as  the 
flanks  of  the  tooth  it  meshes  with.  In  interchangeable  sets  it  is 
desirable  to  use  the  same  size  describing  circle  for  both  the  faces 
and  the  flanks  of  all  the  gears  of  the  same  pitch,  and  the  size  of 
the  describing  circle  which  is  generally  accepted  as  standard  is 
one  whose  diameter  is  equal  to  the  radius  of  a  12-tooth  gear  of  the 
same  pitch.  Here  again,  the  manufacturers  of  gear  cutters  are 
at  variance,  and  use  a  15-tooth  gear  as  the  base  of  the  system. 
This  does  not  mean  that  the  15-tooth  gear  is  the  smallest  gear  in 


290 


GRANT'S  TABLE  FOR  CYCLOIDAL  TEETH     [CHAP.  XII 


the  set,  but  simply  means  that  smaller  gears  will  have  undercut 
flanks. 

219.  Laying  out  the  Cycloidal  Tooth. — The  profiles  of  cycloidal 
teeth,  as  in  the  case  of  involute  teeth,  may  be  very  accurately 
represented  by  circular  arcs.  In  Table  67  are  given  the  radii  of 
these  arcs,  also  the  radial  distances  from  their  centers  to  the  pitch 
line  as  determined  by  Mr.  Grant.  In  laying  out  the  profiles  of 


FIG.  135. 

cycloidal  teeth,  draw  the  pitch,  addendum  and  clearance  circles, 
and  space  off  the  pitch  of  the  teeth  on  the  pitch  circle.  Next 
draw  the  circle  B  as  shown  in  Fig.  135  at  a  distance  a  inside  of  the 
pitch  circle  A,  also  the  circle  C  at  a  distance  e  outside  of  the  pitch 
line.  The  former  is  the  circle  of  face  centers  and  the  latter,  the 

TABLE  67. — RADII  FOR  CYCLOIDAL  TEETH  ACCORDING  TO  G.  B.  GRANT 


Number  of  teeth 

Divide  by  the  diametral  pitch 

Multiply  by  the  circular  pitch 

Exact 

Approx. 

Rad.  b 

Dist.  a 

Rad.  c 

Dist.  e 

Rad.  b 

Dist.  a 

Rad.   c 

Dist.  e 

10 

10 

1.99 

0.02 

-8.00 

4.00 

0.62 

0.01 

-2.55 

1.27 

11 

11 

2.00 

0.04 

-11.05 

6.50 

0.63 

0.01 

-3.34 

2.07 

12 

12 

2.01 

0.06 

CO 

00 

0.64 

0.02 

CO 

00 

13^ 

13-14 

2.04 

0.07 

15.10 

9.43 

0.65 

0.02 

4.80 

3.00 

15^ 

15-16 

2.10 

0.09 

7.86 

3.46 

0.67 

0.03 

2.50 

1.10 

17M 

17-18 

2.14 

0.11 

6.13 

2.20 

0.68 

0.04 

1.95 

0.70 

20 

19-21 

2.20 

0.13 

5.12 

1.57 

0.70 

0.04 

1.63 

€.50 

23 

22-24 

2.26 

0.15 

4.50 

1.13 

0.72 

0.05 

1.43 

0.36 

21 

25-29 

2.33 

0.16 

4.10 

0.96 

0.74 

0.05 

1.30 

0.29 

33 

30-36 

2.40 

0.19 

3.80 

0.72 

0.76 

0.06 

1.20 

0.23 

42 

37-48 

2.48 

0.22 

3.52 

0.63 

0.79 

0.07 

1.12 

0.20 

58 

49-72 

2.60 

0.25 

3.33 

0.54 

0.83 

0.08 

1.06 

0.17 

97 

73-144 

2.83 

0.28 

3.14 

0.44 

0.90 

0.09 

1.00 

0.14 

290 

145-300 

2.92 

0.31 

3.00 

0.38 

0.93 

0.10 

0.95 

0.12 

Rack 

2.96 

0.34 

2.96 

0.34 

0.94 

0.11 

0.94 

0.11 

ART.  220]  CYCLOIDAL  CUTTERS  291 

circle  of  flank  centers.  The  tooth  profile  may  now  be  drawn  us- 
ing the  face  and  flank  radii  6  and  c  given  in  Table  67  for  the  num- 
ber of  teeth  to  be  used  in  the  gear.  The  values  given  for  a,  6, 
c  and  e  in  Table  67  are  for  1  diametrical  pitch  or  1  inch  circular 
pitch.  For  any  other  pitch,  divide  or  multiply  the  tabulated 
values  by  the  given  pitch  as  directed  in  the  table. 

The  smallest  gear  in  the  set  is  again  one  having  ten  teeth,  while 
the  smallest  one  for  which  standard  cutters  are  manufactured  is 
one  having  12  teeth.  The  tooth  form  obtained  by  using  the  tabu- 
lar values  as  directed  above  differs  slightly  from  that  obtained  by 
the  use  of  standard  cutters  on  account  of  the  difference  in  the 
describing  circles,  but  as  in  the  case  of  involutes,  the  discrepancy 
is  small  and  for  that  reason  Grant's  tabular  values  may  be  used 
for  representing  the  tooth  form  on  a  drawing. 

220.  Standard  Cycloidal  Cutters. — The  Brown  and  Sharpe 
Mfg.  Co.  furnish  sets  of  cycloidal  cutters  based  on  the  diametral 
pitch  only,  and  the  sizes  vary  as  follows: 

From  2  to  3  diametral  pitch,  the  pitch  varies  by  quarters. 

From  3  to  4  diametral  pitch,  the  pitch  varies  by  halves. 

From  4  to  10  diametral  pitch,  the  pitch  varies  by  whole  num- 
bers. 

From  10  to  16  diametral  pitch,  the  pitch  varies  by  even  num- 
bers. 

Each  set  consists  of  24  cutters,  as  indicated  in  Table  68. 

TABLE  68. — BROWN  AND  SHARPE  STANDARD  CYCLOIDAL  CUTTERS 
Cutter  A  for  gears  having  12  teeth. 
Cutter  B  for  gears  having  13  teeth. 
Cutter  C  for  gears  having  14  teeth. 
Cutter  D  for  gears  having  15  teeth. 
Cutter  E  for  gears  having  16  teeth. 
Cutter  F  for  gears  having  17  teeth. 
Cutter  G  for  gears  having  18  teeth. 
Cutter  H  for  gears  having  19  teeth. 
Cutter  I  for  gears  having  20  teeth. 
Cutter  J  for  gears  having  21  to  22  teeth. 
Cutter  K  for  gears  having  23  to  24  teeth. 
Cutter  L  for  gears  having  25  to  26  teeth. 
Cutter  M  for  gears  having  27  to  29  teeth. 
Cutter  N  for  gears  having  30  to  33  teeth. 
Cutter  O  for  gears  having  34  to  37  teeth. 
Cutter  P  for  gears  having  38  to  42  teeth. 
Cutter  Q  for  gears  having  43  to  49  teeth. 
Cutter  R  for  gears  having  50  to  59  teeth. 


292 


ACTION  OF  CYCLOIDAL  TEETH 


[CHAP.  XII 


TABLE  68. — BROWN  AND  SHARPE  STANDARD  CYCLOIDAL  CUTTERS. — 

(Continued.) 

Cutter  S  for  gears  having  60  to  74  teeth. 
Cutter  T  for  gears  having  75  to  99  teeth. 
Cutter  U  for  gears  having  100  to  149  teeth. 
Cutter  V  for  gears  having  150  to  249  teeth. 
Cutter  W  for  gears  having  250  or  more. 
Cutter  X  for  gears  having  rack. 

221.  Action  of  Cycloidal  Teeth. — The  action  of  a  pair  of  cy- 
cloidal  teeth  is  illustrated  in  Fig.  136.  Let  the  circles  a  and  6 
represent  the  pitch  circles  of  a  pair  of  gears  having  cycloidal 
teeth,  and  let  the  circles  d  and  e  represent  the  describing  circles. 
Let  C  be  the  pitch  point,  and  Cd  and  Ce  be  the  points  on  the  circles 
d  and  e  which  coincide  with  C  when  the  teeth  are  in  the  position 
shown  in  the  figure.  Now  let  the  centers  of  the  circles  a,  6,  d, 
and  e  be  fixed  and  rotate  a  in  the  direction  indicated  by  the  arrow. 


FIG.  136. 

Let  the  contact  at  C  be  so  arranged  that  the  circles  b,  d,  and  e  are 
driven  with  the  same  peripheral  speed  as  a.  Trace  the  path  of 
the  point  Cd  on  the  surface  of  a  extended,  on  the  surface  of  6 
extended,  and  also  its  actual  path  in  space.  These  paths  will 
evidently  be  the  hypocycloidal  flank  CF,  the  epicycloidal  face 
CH  of  the  meshing  tooth,  and  the  path  of  the  point  of  contact 
CJ.  Now  replace  the  mechanism  in  its  original  position,  rotate 
a  in  the  opposite  direction  and  trace  the  path  of  Ce  in  the  same 
manner.  The  curves  CG,  CI  and  CK,  are  thus  formed  and  they 
complete  the  two  tooth  forms  and  the  path  of  contact.  As  the 
line  of  pressure  between  the  teeth,  which  of  course  coincides  with 
the  common  normal  at.  the  point  of  contact,  must  always  pass 


ART.  221] 


STRENGTH  OF  SPUR  GEARING 


293 


through  the  point  C  in  order  to  transmit  a  uniform  velocity,  the 
angle  of  obliquity  varies  from  the  angle  JCL  to  zero  during  the 
arc  of  approach,  and  from  zero  to  the  angle  KCM,  which  equals 
the  angle  JCL,  during  the  arc  of  recess.  In  order  to  show  that 
with  this  form  of  tooth  the  normal  to  the  tooth  profile  at  the 
point  of  contact  always  passes  through  the  pitch  point  C,  let  us 
study  Fig.  137.  It  is  evident  that  the  generating  point  Ce, 
as  well  as  every  other  point  on  the  rolling  circle,  is  at  any 
given  instant  rotating  about  the  point 
of  contact  C  of  the  rolling  circle 
with  the  pitch  line.  Therefore,  at 
the  instant  in  question  the  line  CCe 
is  a  radius  for  the  point  Ce  and  is  con- 
sequently normal  at  that  point  to 
the  curve  which  Ce  is  generating. 
Now  referring  again  to  Fig.  136,  the 
point  at  which  the  rolling  circle  is 
always  in  contact  with  the  pitch 
circle  is  evidently  the  pitch  point, 
and  therefore  the  common  normal 
at  the  point  of  contact  always 
passes  through  it. 


FIG.  137. 


STRENGTH  OF  SPUR  GEARING 

Having  determined  the  proper  form  of  a  gear  tooth,  the  next 
step  is  to  determine  its  proportions  for  strength.  Owing  to  the 
inaccuracy  of  forming  and  spacing  the  teeth,  it  is  customary  to 
provide  sufficient  strength  for  transmitting  the  entire  load  by  one 
tooth,  rather  than  considering  the  load  as  distributed  over  the 
whole  number  of  teeth  in  theoretical  contact. 

The  load  on  a  single  tooth,  when  the  gears  are  cast  from  wood 
patterns,  is  often  concentrated  at  some  one  point,  usually  an  outer 
corner,  on  account  of  the  draft  on  the  teeth  and  the  natural  warp 
of  the  castings.  The  same  result  is  liable  to  be  produced  when 
the  shaft  is  weak  or  when  the  gears  are  not  supported  on  a  rigid 
framework  or  foundation.  However,  in  the  case  of  well-sup- 
ported machine-moulded  or  cut  gears,  the  load  may  be  considered 
as  uniformly  distributed  along  the  tooth.  For  the  reason  just 
stated,  the  subject  of  the  strength  of  teeth  will  be  discussed  under 
two  heads  as  follows:  (a)  strength  of  cast  teeth;  (6)  strength  of  cut 
teeth. 


294 


STRENGTH  OF  CAST  TEETH 


[CHAP.  XII 


222.  Strength  of  Cast  Teeth.— In  deriving  the  formula  for  the 
maximum  load  that  a  gear  with  cast  teeth  will  transmit,  it  will 
be  sufficiently  accurate  to  consider  the  shape  of  the  tooth  as 
rectangular,  and  the  load  as  acting  at  the  outer  end.  The  load 
may,  however,  be  concentrated  at  one  corner  or  uniformly 
distributed  along  the  length  of  the  tooth. 

(a)  Load  at  one  corner. — With  the  load  concentrated  at  an 
outer  corner  as  shown  in  Fig.  138,  it  is  probable  that  rupture 
would  occur  along  a  section  making  some  angle  a  with  the  base 
of  the  tooth.  Equating  the  bending  moment  about  the  critical 

section  due  to  W  to  the  resist- 
ing moment  of  the  section, 
we  have 

Sht2 


Whcos  a 


6  sin  a 


in  which  S  denotes  the  allow- 
able working  stress  in  the 
material.  From  this  we  get 


S  = 


3TFsin 


(321) 


FIG.  138. 


The    stress    S    is    maximum 
when  sin2o:  is  maximum,  or 
when  a  is  equal  to  45  degrees;  therefore, 


Max.  S  = 


3TF 

t2 


(322) 


(b)  Load  uniformly  distributed. — When  the  load  is  uniformly 
distributed  along  the  length  of  the  tooth,  we  have  by  equating  the 
bending  moment  at  the  base  of  the  tooth  to  the  resisting  moment, 


from  which 


S  = 


ft2 


(323) 


(c)  Equal  strength. — Assuming  that  a  tooth  is  equally  strong 
against  both  methods  of  failure,  the  relation  existing  between 
the  height  h  and  the  face  /  is  found  by  equating  the  stresses  given 
by  (322)  and  (323).  Hence 


ART.  222]  STRENGTH  OF  CAST  TEETH  295 

/  =  2  h  =  1.4  p',  (324) 

where  /i  =  0.7  p'  and  p'  denotes  the  circular  pitch  of  the  gear. 

Although,  as  shown  by  (324),  the  theoretical  length  of  face  at 
which  the  teeth  will  be  of  equal  strength  for  both  cases  of  loading 
is  1.4  p',  a  well-known  American  engineer,  C.  W.  Hunt,  taking 
his  data  from  actual  failures  in  his  own  work,  states  that  the  face 
should  be  about  2  p'  in  order  to  satisfy  this  condition. 

The  seeming  discrepancy  between  theory  and  actual  results 
may  be  easily  explained  when  one  takes  into  consideration  the 
fact  that  even  if  the  load  may  be  entirely  concentrated  at  the 
corner  at  the  beginning  of  application  of  the  load,  it  is  very  pro- 
bable that  before  the  full  pressure  is  brought  to  bear  a  slight  de- 
flection of  the  outer  corner  will  cause  the  load  to  be  distributed 
along  a  considerable  length  of  the  face.  Another  condition 
which  adds  to  the  length  of  the  face  is  that  of  the  proper  propor- 
tions for  wearing  qualities,  and  in  some  cases  the  faces  are  made 
extra  long  for  that  purpose  alone.  It  is  customary  in  American 
practice  to  make  the  face  of  cast  teeth  two  to  three  times  the  cir- 
cular pitch,  the  length  of  the  face  being  increased  as  the  quality 
of  the  work  is  improved. 

(d)  Common  proportions  of  cast  teeth. — The  proportions  of  cast 
gear  teeth  as  used  by  the  different  manufacturers  of  transmission 
gears  vary  somewhat,  but  for  ordinary  service  the  following 
proportions  in  terms  of  the  circular  pitch  have  proven  satisfac- 
tory in  actual  practice: 

Pressure  angle  or  angle  of  obliquity  =  15  degrees. 
Length  of  the  addendum  =  0.3  p'. 
Length  of  the  dedendum  =  0.4  p'. 
Whole  depth  of  the  tooth  =  0.7  p'. 
Working  depth  of  the  tooth  =  0.6  p'. 
Clearance  of  the  tooth  =  0.1  p' 
Width  of  the  tooth  space  =  0.525  p'. 
Thickness  of  the  tooth  =  0.475  p'. 
Backlash  =  0.05  p'. 

(e)  Allowable  working  load  for  cast  teeth. — Assuming  the  pro- 
portions of  the  teeth  as  given  above,  we  find  from  (323)  that  the 
allowable  working  load  on  cast  gear  teeth  has  a  magnitude  given 
by  the  following  expression : 

W  =  0.054  Sp'f  (325) 


296 


STRENGTH  OF  CUT  TEETH 


[CHAP.  XII 


This  formula  has  the  same  general  form  as  the  well-known 
Lewis  formula  given  in  Art.  223.  The  magnitude  of  the  safe 
working  stress  depends  upon  the  material,  the  class  of  service, 
and  the  speed  at  which  the  gears  are  operated.  If  the  gears  are 
subjected  to  heavy  shocks,  due  allowance  must  be  made  for  such 
shocks.  To  obtain  the  probable  safe  working  stress  for  a  given 
speed  and  material,  use  (330)  and  Table  72. 

223.  Strength  of  Cut  Teeth.— In  1893,  Mr.  Wilfred  Lewis  pre- 
sented at  a  meeting  of  the  Engineers'  Club  of  Philadelphia  an 
excellent  method  of  calculating  the  strength  of  cut  gear  teeth. 
His  investigation  was  the  first  one  to  take  into  consideration  the 

form  of  the  tooth  profile  and  the  fact 
that  the  direction  of  the  pressure  is 
always  normal  to  the  tooth  profile. 
The  Lewis  method  has  since  that  time 
been  almost  universally  adopted  for 
calculating  the  strength  of  teeth  when 
the  workmanship  is  of  high  grade,  as  in 
the  cut  gears,  and  not  infrequently 
for  machine-moulded  teeth. 

In  this  investigation,  Mr.  Lewis  as- 
sumed that  at  the  beginning  of  contact 
the  load  was  concentrated  at  the  end  of 
the  tooth,  with  its  line  of  action  normal 
to  the  tooth  profile  in  the  direction  AB 
as  shown  in  Fig.  139.  The  actual  thrust  P  was  then  resolved  at 
the  point  B  into  two  components,  one  acting  radially  producing 
pure  compression,  and  the  other,  W,  acting  tangentially.  When 
the  material  of  which  the  gears  are  made  is  stronger  in  compres- 
sion than  in  tension,  the  radial  component  adds  to  the  strength 
of  the  tooth,  and  when  the  tensile  and  compressive  strengths  are 
approximately  equal,  it  is  a  source  of  weakness.  However,  in 
either  case  the  effect  is  not  marked,  and  in  the  original  investiga- 
tion was  neglected  altogether. 

The  strength  of  the  tooth  may  now  be  determined  by  drawing 
through  the  point  B}  Fig.  139,  a  parabola  which  is  tangent  to 
the  tooth  profile  at  the  points  D  and  E.  This  parabola  then  en- 
closes a  cantilever  beam  of  uniform  strength  as  the  following 
analysis  shows. 

A  beam  of  uniform  strength  is  one  in  which  the  fiber  stress  due 
to  bending  is  constant.  For  the  case  under  discussion,  by  equat- 


FIG.  139. 


ART.  223] 


TABLE  OF  LEWIS  FACTORS 


297 


ing  the  external  moment  to  the  moment  of  resistance,  we  obtain 

(326) 


Wh  =  ^> 


from  which 


(327) 


thus  proving  that  a  beam  of  uniform  strength  has  a  parabolic 
outline. 

Since  the  actual  tooth  and  the  inscribed  parabola  have  the 
same  value  of  t  as  shown  in  Fig.  139,  it  is  evident  that  the  para- 
bolic beam  must  be  a  measure  of  the  strength  of  the  gear  tooth, 
and  that  the  weakest  section  of  the  tooth  must  lie  along  DE. 

The  problem  now  is  to  find  an  expression  for  the  load  W  in 
terms  of  the  dimensions  of  the  tooth,  the  safe  fiber  stress  and  a 
constant.  From  the  similar  triangles  shown  in  Fig.  139,  it  fol- 
lows that 

t2  =  4  hx  (328) 

Combining  (326)  and  (328),  we  find 

W  =  %  Sfx 
TABLE  69. — LEWIS  FACTORS  FOR  GEARING 


No.  of 
teeth 

Involute 

Radial 

flank 

Cycloid 

No.  of 
teeth 

Involute 

Radial 
flank 

Cycloid 

15° 

20° 

15° 

20° 

12 

0.067 

0.0780 

0.0520 

40 

0.1070 

0.1312 

0.0674 

13 

0.071 

0.0840 

0.0530 

45 

0.1080 

0.1340 

0.0682 

14 

0.075 

0.0890 

0.0540 

50 

0.1100 

0.1360 

0.0690 

15 

0.078 

0.0930 

0.0550 

55 

0.1120 

0.1375 

16 

0.081 

0.0970 

0.0560 

60 

0.1130 

0.1390 

0.0700 

17 

0.084 

0.1000 

0.0570 

Same 

65 

0.1140 

0.1400 

Same 

18 

0.086 

0.1030 

0.0580 

values 

70 

0.1144 

0.1410 

values 

19 

0.088 

0.1060 

0.0590 

as 

75 

0.1150 

0.1420 

0.0710 

as 

20 

0.090 

0.1080 

0.0600 

for 

80 

0.1155 

0.1426 

for 

21 

0.092 

0.1110 

0.0610 

15° 

90 

0.1164 

0  .  1440 

15° 

22 

0.093 

0.1130 

0.0615 

invo- 

100 

0.1170 

0.1450 

0.0720 

invo- 

23 

0.094 

0.1140 

0.0620 

lute 

120 

0.1180 

0.1460 

lute 

24 

0.096 

0.1160 

0.0625 

140 

0.1190 

0.1475 

26 

0.098 

0.1190 

0.0635 

160 

0.1197 

0.1483 

28 

0.100 

0.1220 

0.0643 

180 

0.1202 

0.1490 

30 

0.101 

0.1240 

0.0650 

200 

0.1206 

0.1495 

0.0730 

33 

0.103 

0.1260 

0.0657 

250 

0.1213 

0.1504 

36 

0.105 

0.1290 

0.0665 

300 

0.1217 

0.1510 

0.0740 

39 

0.107 

0.1306 

0.0672 

Rack 

0.1240 

0.1540 

0.0750 

298 


LEWIS  FACTORS  FOR  STUB-TEETH         [CHAP.  XII 


Dividing  and  multiplying  by  p',  the  circular  pitch, 

W  =  Sp'fy,  (329) 

2  x 

in  which  y  =  ~ — ,  is  a  factor  depending  upon  the  pitch  and  form 
6p 

of  the  tooth  profile.  The  value  of  this  factor  must  be  obtained 
from  a  layout  of  the  tooth,  provided  a  table  of  such  factors  is  not 
available.  For  convenience,  the  factor  y  will  hereafter  be  known 
as  the  "Lewis  factor"  and  in  Table  69  are  given  the  values  of  this 

TABLE  70. — VALUES   OF   y  IN  LEWIS'  FORMULA  FOR  STUB-TOOTH  GEARS. 


No.  of 
teeth 

Fellows  system 

Nuttall 
system 

% 

H 

H 

K 

Sio 

Hi 

»Ha 

1#4 

12 

0.096 

0.111 

0.102 

0.100 

0.096 

0.100 

0.093 

0.092 

0.099 

13 

0.101 

0.115 

0.107 

0.106 

0.101 

0.104 

0.098 

0.096 

0.103 

14 

0.105 

0.119 

0.112 

0.111 

0.106 

0.108 

0.102 

0.100 

0.108 

15 

0.108 

0.123 

0.115 

0.115 

0.110 

0.111 

0.105 

0.103 

0.111 

16 

0.111 

0.126 

0.119 

0.118 

0.113 

0.114 

0.109 

0.106 

0.115 

17 

0.114 

0.129 

0.122 

0.121 

0.116 

0.116 

0.111 

0.109 

0.117 

18 

0.117 

0.131 

0.124 

0.124 

0.119 

0.119 

0.114 

0.111 

0.120 

19 

0.119 

0.133 

0.127 

0.127 

0.122 

0.121 

0.116 

0.113 

0.123 

20 

0.121 

0.135 

0.129 

0.129 

0.124 

0.123 

0.118 

0.115 

0.125 

21 

0.123 

0.137 

0.131 

0.131 

0.126 

0.125 

0.120 

0.117 

0.127 

22 

0.125 

0.139 

0.133 

0.133 

0.128 

0.126 

0.122 

0.118 

0.128 

23 

0.126 

0.141 

0.134 

0.135 

0.129 

0.128 

0.123 

0.120 

0.130 

24 

0.128 

0.142 

0.136 

0.136 

0.131 

0.129 

0.125 

0.121 

0.131 

25 

0.129 

0.143 

0.137 

0.138 

0.133 

0.130 

0.126 

0.123 

0.133 

26 

0.130 

0.145 

0.139 

0.139 

0.134 

0.132 

0.128 

0.124 

0.134 

27 

0.132 

0.146 

0.140 

0.140 

0.135 

0.133 

0.129 

0.125 

0.136 

28 

0.133 

0.147 

0.141 

0.141 

0.136 

0.134 

0.130 

0.126 

0.137 

29 

0.134 

0.148 

0.142 

0.143 

0.137 

0.135 

0.131 

0.127 

0.138 

30 

0.135 

0.149 

0.143 

0.144 

0.138 

0.136 

0.132 

0.128 

0.139 

32 

0.137 

0.150 

0.145 

0.146 

0.140 

0.137 

0.134 

0.130 

0.141 

35 

0.139 

0.153 

0.147 

0.148 

0.143 

0.139 

0.136 

0.132 

0.143 

37 

0.140 

0.154 

0.149 

0.149 

0.144 

0.141 

0.138 

0.133 

0.145 

40 

0.142 

0.156 

0.151 

0.151 

0.146 

0.142 

0.140 

0.135 

0.146 

45 

0.145 

0.159 

0.154 

0.154 

0.149 

0.145 

0.142 

0.138 

0.149 

50 

0.147 

0.161 

0.156 

0.156 

0.151 

0.147 

0.144 

0.140 

0.151 

55 

0.149 

0.162 

0.157 

0.158 

0.152 

0.149 

0.146 

0.141 

0.153 

60 

0.150 

0.164 

0.159 

0.159 

0.154 

0.150 

0.148 

0.143 

0.154 

70 

0.153 

0.166 

0.161 

0.161 

0.156 

0.152 

0.150 

0.145 

0.157 

80 

0.155 

0.168 

0.163 

0.163 

01.58 

0.154 

0.152 

0.147 

0.159 

100 

0.158 

0.171 

0.166 

0.166 

0.160 

0.156 

0.154 

0.150 

0.161 

150 

0.162 

0.174 

0.170 

0.169 

0.164 

0.160 

0.158 

0.154 

0.165 

200 

0.164 

0.176 

0.172 

0.171 

0.166 

0.162 

0.160 

0.156 

0.167 

Rack 

0.173 

0.184 

0.179 

0.176 

0.172 

0.170 

0.168 

0.166 

0.175 

ART.  224] 


PROPORTIONS  OF  CUT  TEETH 


299 


factor  as  worked  out  by  Mr.  Lewis  for  the  several  systems  of  gear- 
ing. In  Table  70  are  given  the  values  of  the  Lewis  factor 
for  the  two  systems  of  stub-tooth  gearing  in  common  use. 
These  factors  were  derived  and  tabulated  by  Mr.  L.  G.  Smith 
under  the  direction  of  the  author,  and  formed  a  part  of  a  thesis 
submitted  by  Mr.  Smith. 

(b)  Proportions  of  cut  teeth. — The  proportions  of  cut  teeth  as 
recommended  by  several  manufacturers  of  gear-cutting  machin- 
ery vary  considerably,  as  may  be  noticed  from  an  inspection  of  the 
formulas  given  in  Table  71.  No  doubt  the  formulas  proposed 
by  the  Brown  and  Sharpe  Co.  for  the  common  system  of  gearing 
are  used  more  extensively  than  any  other  and  are  generally 
recognized  as  the  standard.  The  formulas  due  to  Hunt  apply  to 
short  teeth,  while  those  proposed  by  Messrs.  Logue  and  Fellows 
apply  to  the  well-known  stub  systems  of  gear  teeth.  No  for- 
mulas are  given  in  Table  71  for  the  Fellows  stub  teeth  since  this 
system  is  discussed  more  in  detail  in  Art.  230  (d).  It  should  be 
noted  that  the  proportions  recommended  by  Messrs.  Hunt  and 
Logue  agree  on  all  points  except  the  pressure  angle. 

TABLE  71. — PROPORTIONS  OF  CUT  TEETH 


Brown  and 
Sharpe 

Hunt 

Logue 

Fellows 

Pressure  angle 

14K° 

14  M° 

20° 

20° 

Length  of  addendum  

0.3183  p' 

0.25p' 

0  .  25  p' 

Length  of  dedendum  

0  3683  p' 

0  30  p' 

0  30  pr 

Whole  depth  of  tooth 

0  6866  p' 

0  55  p' 

0  55  pr 

Working  depth  of  tooth  
Clearance  

0.6366p' 

O.SOp' 
0  05  p' 

0.50p' 

Width  of  tooth  space 

0  50  p' 

Thickness  of  the  tooth  
Backlash  

0.50?/ 
0 

Another  important  fact  shown  in  the  table  is  that  for  cut  teeth 
the  backlash  is  zero. 

224.  Materials  and  Safe  Working  Stresses. — (a)  Materials 
used  in  gears. — The  factor  S  in  the  Lewis  formula  depends  upon 
the  material  used  in  the  construction  of  the  gears.  The  materials 
used  for  gear  teeth  are  various  grades  of  alloy  steels,  machine 
steel,  steel  casting,  semi-steel,  cast  iron,  bronze,  rawhide,  cloth, 
fiber,  and  wood.  Machine-steel  pinions  are  used  with  large  cast- 
iron  gears;  the  use  of  the  stronger  material  makes  up  for  the 


300  MATERIALS  FOR  GEARS  [CHAP.  XII 

weakness  of  the  teeth  on  the  pinion,  due  to  the  decreased  section 
at  the  root.  At  the  present  time  the  majority  of  the  gears  used 
in  motor-car  construction  are  made  of  steel  and  are  then  sub- 
jected to  a  heat  treatment,  the  effect  of  which  has  been  discussed 
in  Arts.  52  and  53.  Many  gears  on  modern  machine  tools  and 
electric  railway  cars  are  made  of  steel  and  then  given  a  heat 
treatment.  Steel  gears  heat  treated  are  stronger  and  are 
capable  of  resisting  wear  much  better  than  untreated  gears. 

Steel  casting  is  used  when  the  gears  are  of  large  size.  This 
material  is  well  adapted  for  resisting  shocks  and,  being  much 
stronger  than  cast  iron,  it  is  used  for  service  in  which  heavy  loads 
prevail.  Semi-steel,  which  is  nothing  more  than  a  high-grade 
cast  iron,  is  also  used  for  large  gears  where  the  shocks  and  loads 
are  not  so  severe.  Cast  iron  probably  is  used  more  frequently 
than  any  other  material,  and  in  many  cases  the  manufacturers 
of  gears  use  a  special  cupola  mixture  that  will  produce  a  tough 
and  close-grained  metal. 

Bronze  is  frequently  used  for  spur  pinions  meshing  with  steel 
or  iron  gears,  and  when  the  teeth  are  properly  cut  the  gears 
may  be  run  at  fairly  high  speeds.  In  worm-gear  installations, 
the  gear  is  generally  made  of  bronze  and  the  worm  of  a  good  grade 
of  steel,  in  many  cases  heat  treated.  Several  manufacturers  are 
now  making  special  gear  bronzes  that  are  adapted  for  a  particular 
type  of  service.  Some  of  these  bronzes  are  discussed  more  or  less 
fully  in  the  chapter  on  worm  gearing.  In  general,  bronze  is 
much  stronger  than  ordinary  cast  iron  when  applied  to  gear  teeth. 

Rawhide,  cloth,  and  fiber  gears  are  used  when  quiet  and  smooth- 
running  gears,  free  from  vibration,  are  desired.  Rawhide  gears 
are  stronger  and  are  preferable  to  ordinary  fiber  ones.  The 
New  Process  Rawhide  Co.  claims  its  gears  to  be  equally  as  strong 
as  cast-iron  gears  of  the  same  dimensions.  Such  gears  are  fur- 
nished with  or  without  metal  flanges  and  bushings,  and  the  teeth 
are  cut  the  same  as  in  a  metal  gear.  As  ordinarily  constructed, 
the  flanges  and  hub  of  the  smaller  gears  are  made  of  brass  or 
bronze  and  for  the  larger  ones  cast  iron  or  steel  may  be  used. 
In  the  case  of  large  gears  only  the  teeth  and  rim  are  of  rawhide, 
the  center  being  of  cast  iron.  As  a  rule,  however,  rawhide  gears 
are  of  small  size.  They  are  often  used  as  the  driving  pinions 
on  motor  drives,  and  the  fact  that  rawhide  is  a  non-conductor  is 
in  this  service  a  marked  advantage. 

The  cloth  or  so-called   "Fabroif"   gears  introduced  several 


ART.  224]  SAFE  WORKING  STRESS  301 

years  ago  by  the  General  Electric  Co.  consist  of  a  filler  of  cotton 
or  similar  material  confined  at  a  high  pressure  between  steel 
flanges  held  together  by  either  threaded  rivets  or  sleeves,  depend- 
ing upon  the  size  of  the  gears.  After  cutting  the  teeth  in  the 
blank,  the  gear  is  subjected  to  an  oil  treatment  making  it  mois- 
ture-proof as  well  as  vermin-proof.  In  strength,  Fabroil  gears 
are  the  equal  of  other  non-metallic  gears,  and  according  to  the 
manufacturer  they  may  be  used  in  practically  any  service  where 
cast  iron  gears  are  used. 

Recently  the  Westinghouse  Electric  and  Manufacturing  Co. 
placed  upon  the  market  a  non-metallic  or  fibrous  material,  called 
Bakelite  Micarta-D,  that  is  suitable  for  gears  and  pinions.  It 
is  especially  adapted  for  installations  where  it  is  desirable  to 
transmit  power  with  a  minimum  amount  of  noise.  This  material 
possesses  good  wearing  qualities,  is  vermin-proof,  absorbs  practi- 
cally no  oil  or  water,  and  is  unaffected  by  atmospheric  changes 
and  acid  fumes.  Furthermore,  gears  made  of  this  material  may 
be  run  in  oil  without  showing  any  signs  of  injury;  in  fact,  the 
manufacturers  specify  that  a  good  lubricating  oil  or  grease  is 
essential  in  order  to  obtain  good  results.  According  to  recorded 
tests,  the  ultimate  tensile  strength  of  Bakelite  Micarta-D  is 
approximately  18,000  pounds  per  square  inch  with  the  grain, 
while  its  compressive  strength  across  the  grain  is  40,000  pounds 
per  square  inch. 

(6)  Safe  working  stress.  —  The  factor  S  in  (329)  depends  upon 
the  kind  of  material  used,  the  conditions  under  which  the  gears 
run  and  the  velocity  of  the  gears.  If  the  gears  are  subjected  to 
severe  fluctuations  of  load  or  to  shock,  or  both,  due  allowance 
must  be  made.  To  provide  against  the  effect  of  speed,  Mr. 
Lewis  published  a  table  of  allowable  working  stresses  for  a  few 
types  of  materials.  Some  years  later  Mr.  C.  G.  Barth  originated 
an  equation  giving  values  for  S  which  agree  very  closely  with 
those  recommended  by  Mr.  Lewis.  The  Barth  formula  is  gener- 
ally put  into  the  following  form  : 


in  which  So  denotes  the  permissible  fiber  stress  of  the  material  at 
zero  speed  and  V  the  pitch  line  velocity  in  feet  per  minute.  In 
Table  72  are  given  values  of  So  for  the  various  materials 
discussed. 


302  TABLE  OF  VALUES  SQ  [CHAP.  XII 

TABLE  72. — VALUES  OF  So  FOR  VARIOUS  MATERIALS 


Materials 

So 

1 

Chrome  nickel  steel,  hardened     .       

100,000 

2 

Chrome  vanadium  steel,  hardened 

100,000 

3 

Alloy  steel,  case-hardened  

50,000 

4 

Machinery  steel  .          

25,000 

5 

Steel  casting  . 

20,000 

6 

Special  high-grade  bronze 

16  000 

7 

Ordinary  bronze  .  .              

12,000 

8 

High-grade  cast  iron  (semi-steel)   .                   .  . 

15,000 

9 

Good  cast  iron  

10,000 

10 

Ordinary  cast  iron           .                      

8,000 

11 

Fab  roil                                                                                .  .  . 

8,000 

12 

Bakelite  Micarta-D 

8,000 

13 

Rawhide  ...                                             

8,000 

GEAR  CONSTRUCTION 

The  constructive  details  of  gears  depend  largely  upon  the  size, 
and  to  some  extent  upon  the  material  used  as  well  as  upon  the 
machine  part  to  which  the  gears  are  fastened.  Small  metal 
gears  are  generally  made  solid,  but  when  the  diameter  gets  too 
large  for  this  type  of  construction  thus  producing  a  heavy  gear, 
the  weight  of  such  gears  can  be  materially  decreased  by  recessing 
the  sides  thus  forming  a  central  web  connection  between  the  rim 
and  the  hub.  Not  infrequently  round  holes  are  put  through  the 
web,  thus  effecting  an  additional  saving  in  weight  and  at  the  same 
time  giving  the  gear  an  appearance  of  having  arms.  Gear  blanks 
having  a  central  web  are  usually  produced  by  casting,  or  by  a 
drop  forging  operation. 

225.  Rawhide  Gears. — Rawhide  gears,  as  mentioned  in  the  pre- 
ceding article,  are  always  provided  with  metal  flanges  at  the  side 
as  illustrated  in  the  various  designs  shown  in  Figs.  140  and  141. 
For  spur  gears  up  to  and  including  9  inches  outside  diameter,  the 
metal  flanges  are  fastened  together  by  means  of  rivets  with  coun- 
tersunk heads  as  shown.  For  larger  outside  diameters  either 
rivets  or  through  bolts  are  used,  depending  largely  upon  sur- 
rounding conditions. 

The  design  shown  in  Fig.  140 (a),  having  the  plates  extending 
almost  to  the  roots  of  the  teeth,  produces  a  very  quiet  running 
gear  which  gives  good  service  for  light  and  medium  loads.  In  this 


ART.  225] 


RAWHIDE  GEARS 


303 


case  the  flanges  are  merely  used  for  supporting  the  key  If  a 
stronger  rawhide  gear  is  desired  than  that  just  described,  the 
flanges  must  be  extended  to  the  ends  of  the  teeth,  thus  forming  the 
combination  shown  in  Fig.  140(6).  The  flanges  may  or  may  not 


(a) 


(b) 

FIG.  140. 


(c) 


form  a  part  of  the  working  face.  If  the  working  face  does  not 
include  the  flanges,  the  rawhide  filler  must  be  made  Y±  inch  wider 
than  the  face  of  the  engaging  gear;  furthermore,  if  this  gear  is 
used  as  a  motor  pinion,  the  rawhide  face  must  be  considerably 


Co) 


(b) 


FIG.   141. 


wider  than  the  face  of  the  mating  gear  in  order  to  compensate 
for  the  floating  of  the  armature  shaft.  The  object  of  extending 
the  flanges  to  the  tops  of  the  teeth  is  to  prevent  the  outer  layers 
of  rawhide  from  curling  over  and  thus  eventually  ruining  the 
whole  gear. 


304 


RAWHIDE  GEARS 


[CHAP.  XII 


The  design  shown  in  Fig.  140 (c)  is  intended  for  severe  service. 
Quiet  operation  is  obtained  by  eliminating  the  metal  to  metal 
contact,  and  this  is  accomplished  by  making  the  rawhide  face 
somewhat  wider  than  the  face  of  the  engaging  gear.  The  con- 
struction shown  in  Fig.  141  (a)  is  that  used  for  large  gears,  thus 
effecting  a  considerable  saving  of  rawhide  by  using  the  cast-iron 
spider  to  which  the  rawhide  rim  is  fastened  as  shown.  The 
flanges  may  or  may  not  extend  to  the  tops  of  the  teeth.  When 
the  face  of  such  a  gear  is  4  inches  or  more,  through  bolts  are 
generally  used  in  place  of  rivets,  unless  the  projecting  heads  and 
nuts  are  found  objectionable. 

For  the  constructions  shown  in  Fig.  140(a)  and  (6),  the  thick- 
ness of  the  plates  may  be  made  according  to  the  dimensions  given 
in  Table  73.  This  table  also  gives  the  size  of  rivets  to  be  used 
for  a  given  pitch  of  tooth  and  for  the  ordinary  length  of  face, 
namely,  about  three  times  the  circular  pitch.  The  last  two  col- 
umns given  in  the  table  refer  to  the  minimum  radial  thickness  of 
the  rawhide  blank  when  used  without  and  with  a  metal  spider. 

TABLE  73. — DATA  PERTAINING  TO  RAWHIDE  GEARS 


Diametral  pitch 

Flange 
'  thickness 

Diameter  of 
rivet 

Thickness  of  rawhide  rim 

Without  metal 
spider 

With  metal 
spider 

12 
10 

9 

8 
7 
6 
5 
4 
3 
2% 
2K 

2^ 
2 

l/8 

%2 

0.445 
0.550 
0.590 
0.640 
0.725 
0.890 
1.100 
1.275 
1.675 
1.780 
1.905 
2.140 
2.330 

0.415 
0.505 
0.545 
0.590 
0.670 
0.800 
0.975 
1.150 
1.500 
1.610 
1.735 
1.980 
2.175 

Ys 

H 

%4 

He 

K 

H 

Me 

y* 

H 

The  information  included  in  Table  73  was  kindly  furnished  by 
the  New  Process  Gear  Corporation  and  represents  their  practice 
in  the  ordinary  designs  of  rawhide  gears. 


ART.  226] 


FABROIL  GEARS 


305 


226.  Fabroil  Gears. — The  general  constructive  features  of 
Fabroil  gears  are  very  much  the  same  as  those  used  for  rawhide 
gears.  In  the  usual  construction  as  recommended  by  the  General 
Electric  Co.,  the  flanges  are  made  of  steel  and  threaded  studs  are 


(a) 


(b) 


FIG.  142. 


used  for  clamping  the  flanges  together.  Four  different  designs 
of  such  gears  are  shown  in  Figs.  142  and  143.  The  first  of  these, 
Fig.  142 (a),  shows  the  standard  construction  without  a  metal 
bushing  or  spider.  For  gears  of  a  larger  size,  a  flanged  bushing 


(b) 


FIG.  143. 


made  of  machine  steel  or  steel  casting,  depending  upon  the  size 
of  the  gear,  is  employed,  as  shown  in  Fig.  142(6).  This  form  of 
construction  is  used  for  the  sake  of  economy  of  material. 
The  two  flanges  are  locked  together  by  the  threaded  studs, 


306  BAKELITE  MICARTA-D  GEARS  [CHAP.  XII 

and  in  addition  the  removable  flange  is  locked  to  the  bushing  by 
several  studs  tapped  half  into  the  flange  and  half  into  the  cen- 
tral bushing. 

The  design  shown  in  Fig.  143  (a)  represents  the  construction 
used  for  large  gears  when  it  is  desirable  to  save  material.  The 
Fabroil  rim  is  pressed  over  a  metal  spider  and  locked  in  place 
by  the  system  of  threaded  studs  shown  in  the  figure.  The 
threaded  sleeve  construction  shown  in  Fig.  143(6)  is  used  for 
small  pinions  where  there  is  not  sufficient  room  for  the  threaded 
studs  used  in  the  standard  construction.  The  end  flanges  are 
locked  to  the  threaded  sleeve  in  the  manner  shown  in  the  figure. 

227.  Bakelite   Micarta-D   Gears. — For   ordinary   service,   in 
which  the  face  of  the  Bakelite  Micarta-D  is  made  equal  to  or  less 
than  the  face  of  the  mating  gear,  no  end  flanges  are  required  since 
the  material  is  self-supporting.     However,  end  flanges  are  recom- 
mended when  it  is  desired  to  transmit  heavy  loads  or  when  the 
diameter  of  the  gear  is  more  than  four  times  its  face.     Bakelite 
Micarta-D  is  obtainable  in  the  form  of  plates  up  to  36  inches 
square  and  in  thicknesses  varying  from  KG  mcn  "to  2  inches; 
hence  the  largest  gears  that  can  be  made  are  limited  to  36  inches 
outside  diameter,  but  the  face  may  be  made  any  width  whatsoever 
by  riveting  together  two  or  more  plates.     For  economy  of  mate- 
rial, large-diameter  gears  are  made  with  a  metal  center  similar  to 
that  of  the  Fabroil  gear  shown  in  Fig.  143 (a). 

228.  Large  Gears. — Gears  of  medium  diameter  are  cast  in  one 
piece,  either  of  cast  iron  or  of  steel  casting  depending  upon  the 
class  of  service  for  which  they  are  intended.     Quite  often  the 
gear  is  cast  in  one  piece  and  in  order  to  relieve  the  shrinkage 
stresses,  due  to  excessive  metal  in  the  hub,  the  latter  is  split  and 
the  halves  are  then  utilized  for  clamping  the  gear  to  the  shaft. 
Such  a  gear  is  shown  in  Fig.  178  of  Chapter  XIV.     Frequently 
it  is  desirable  to  use  a  split  construction,  by  which  is  meant  the 
gear  is  made  in  two  halves  that  are  bolted  together.     This  is  a 
common  form  of  construction  in  railway  motor  gears,  and  as 
usually  made,  the  joint  comes  along  an  arm. 

In  Fig.  144  is  shown  the  design  of  a  triple-staggered-tooth  spur 
gear  designed  and  constructed  by  the  Mesta  Machine  Co.  of 
Pittsburg.  The  design  is  quite  different  from  that  ordinarily 
used,  in  that  the  face  of  the  gear  is  built  up  of  three  separate  rims 
bolted  together  with  the  teeth  in  the  three  sections  arranged  in  a 


ART.  229] 


LARGE  GEARS 


307 


staggered  order.  The  gear  is  actually  constructed  of  six  sepa- 
rate parts.  The  central  part  or  spider  is  split  through  the  hub 
and  rim  between  two  arms  and  has  bolted  to  it  the  separate 
outside  rims,  each  of  which  consists  of  two  halves.  The  gear  is 
used  for  driving  a  sheet  mill  and  is  capable  of  transmitting  1,600 
horse  power  at  a  pitch  line  speed  of  2,000  feet  per  minute.  The 
gear  contains  154  teeth  of  5j-^-inch  circular  pitch  and  has  a  face 
of  38  inches,  and  the  pinion  driving  the  gear  has  20  teeth. 
Because  of  the  high  pitch  line  speed  the  drive  is  arranged  to  run 
in  an  oil  bath. 


FIG.  144. 

Another  interesting  design  of  a  large  gear  is  that  in  which  a 
separate  spider,  consisting  of  hub  and  arms,  has  bolted  to  it  a 
rim  built  up  in  sections.  For  an  illustration  of  this  type,  as  well 
as  other  designs  of  large  gears,  consult  Chapter  XIV. 

229.  Gear-wheel  Proportions. — (a)  Arms. — An  exact  analysis 
of  the  stresses  produced  in  gear  arms  is  exceedingly  difficult,  and 
as  far  as  the  author  is  aware,  no  such  analysis  has  ever  been  pre- 
sented. In  arriving  at  a  formula  by  means  of  which  the  dimen- 
sions of  the  arm  may  be  calculated,  we  shall  assume  that  the  rim 


308 


GEAR  PROPORTIONS 


[CHAP.  XII 


is  of  sufficient  thickness  that  the  load  on  the  teeth  is  distrib- 
uted equally  among  the  arms.  No  doubt  this  assumption  is 
justifiable,  since  the  rim  must  be  made  so  rigid  that  it  is  subjected 
to  no  bending  between  the  arms. 


/^// 


A~VJLN>-'A 


(b) 


Fid.  145. 


With  this  assumption,  we  get  the  following  expression  for  the 
section  modulus  of  the  arm  at  the  center  of  the  hub: 


WR 


(331) 


FIG.  146. 


in  which  W  is  the  load  on  the  gear;  R  the  length  of  the  arm  or,  in' 
this  case,  the  radius  of  the  gear  in  inches;  S  the  allowable  fiber 
stress  in  the  material,  and  n  the  number  of  arms  in  the  gear. 

By  means  of  (331),  the  value  of  the  section  modulus  -  may  be 


ART.  229] 


GEAR  PROPORTIONS 


309 


determined,  from  which  the  dimensions  of  the  adopted  arm  sec- 
tion may  be  obtained.  In  Figs.  145  and  146  are  shown  four  types 
of  arms  that  are  used  in  gear  construction.  Of  these,  the  de- 
signs shown  by  Fig.  145(6)  and  Fig.  146  (a)  and  (6)  are  used 
chiefly  for  large  and  heavy  gears,  while  the  elliptical  arm  shown 
in  Fig.  145  (a)  is  intended  for  lighter  service,  although  very  often 
it  is  also  used  for  heavy  work.  The  proportions  of  the  various 
arm  sections  illustrated  in  the  above  figures  are  given  in  Fig.  147. 
The  dimensions  of  the  arm  at  the  pitch  line  are  generally  made 
approximately  seven-tenths  of  those  at  the  center.  Since  the 
elliptical  cross-section  is  used  largely  for  the  ordinary  gears,  we 
shall  derive  a  formula  for  that  section  assuming  the  proportions 


vsSSS^  $ssSSss«l__Z 


(b)  (c) 

FIG.  147. 


given  in  Fig.  147  (a).     The  section  modulus  for  an  ellipse,  having 

TT/l3 

the  proportions  referred  to  above,  is  -r\  hence  from  (331),  we 


get 


h  = 


(332) 


The  number  of  arms  in  gears  varies  with  the  diameter,  and  the 
following  represents  the  prevailing  practice: 

1.  Four  or  five  arms  for  gears  up  to  16  or  20  inches  in  diameter. 

2.  Six  arms  for  gears  from  16  to  60  inches  in  diameter. 

3.  Eight  arms  for  gears  from  60  to  96  inches  in  diameter. 

4.  Ten  or  twelve  arms  for  gears  above  96  inches  in  diameter. 
Web  centers  are  used  for  smaller  gears,  and  the  thickness  of  the 

web  approximates  one-half  of  the  circular  pitch.  Sometimes 
stiffening  ribs  are  introduced  between  the  hub  and  rim,  and  the 
thickness  of  such  ribs  is  generally  equal  to  the  web  thickness. 

(6)  Rim. — Calculations  for  the  rim  dimensions  are  of  little 
value,  and  in  actual  designing  empirical  formulas  are  resorted  to. 


310 


GEAR  PROPORTIONS 


[CHAP.  XII 


As  shown  in  Figs.  145  and  146,  the  minimum  thickness  of  the  rim 
under  the  teeth  is  made  about  one-half  the  circular  pitch  and 
should  taper  to  a  slightly  greater  thickness  where  the  arms  join 
the  rim.  Good  design  dictates  that  the  rim  should  be  supplied 
with  a  central  rib  or  bead,  as  illustrated  in  Figs.  145  and  146. 

(c)  Face  of  gears. — The  width  of  the  face  of  a  gear  depends  in 
general  upon  the  type  of  gear,  whether  it  has  cast  or  cut  teeth, 
the  class  of  service,  and  the  location  of  the  gear.     If  the  gear  is 
located  between  rigid  bearings,  the  face  may  be  made  wider  than 
when  the  gear  is  a  considerable  distance  from  the  bearings,  since 
in  the  latter  case  the  deflection  of  the  shaft  due  to  the  load  on  the 
gear  might  seriously  affect  the  distribution  of  the  load  across  a 
wide  face.     For  cast  teeth  it  is  good  practice  to  make  the  face 
from  two  to  three  times  the  circular  pitch,  while  for  cut  teeth 
the  face  is  made  from  two  and  one-half  to  six  times  the  circular 
pitch,  three  to  four  being  a  fairly  good  average. 

(d)  Hubs. — The  hubs  of  gears  are  made  either  solid  or  split,  as 
stated  in  the  preceding  article.     In  either  type  of  hub  good  design 
calls    for    a    reinforcement    of   metal    over  the  key,   and   this 
condition   is   met   if  the   hubs   are   proportioned  according  to 
suggestions  offered  in  Figs.  145  and  146.     The  object  of  a  split 
hub  is  to  reduce  the  cooling  stresses  in  the  gear  and  at  the 
same  time  permit  any  desired  adjustment  of  the  gear  on  the 
shaft.     Keys  should  always  be  placed  under  an  arm  in  the  case  of 
a  solid  hub,  and  in  a  split  hub  approximately  at  right  angles  to 
the  center  split  or  hub  joint.     The  diameters  and  lengths  of  the 
hub  may  be  made  in  accordance  with  the  formulas  given  in  Table 
74,  in  which  d  denotes  the  bore  of  the  hub.     These  formulas 
were  published  by  Herman  Johnson  in  the  American  Machinist 
of  Jan.  14,  1904,  and  represent  the  actual  practice  of  four  large 
manufacturers. 


TABLE  74. — DIMENSIONS  OF  GEAR  HUBS 


Type  of  service 

Diameter 

Length 

Cast  iron 

Steel  casting 

Heavy  load  and  great  shock 
Medium   load  and   medium 
shock  

2d 
1.75d+0.125" 

1.625  d+0.125" 

1.75d+0.125" 
1.  625  d  +0.1875" 

1.5d+0.25 

Light  load  and  no  shock  

ART.  230] 


METHODS  OF  STRENGTHENING 


311 


230.  Methods  of  Strengthening  Gear  Teeth.— Occasionally  it 
is  desirable  to  have  the  teeth  of  a  gear  extra  strong,  and  to  obtain 
additional  strength  any  one  of  the  seven  following  methods  may 
be  used:  (a)  shrouding;  (b)  use  of  short  teeth;  (c)  increase  of  the 
angle  of  obliquity;  (d)  use  of  stub  teeth;  (e)  use  of  unequal  adden- 
dum teeth;  (/)  use  of  buttressed  teeth;  (g)  use  of  helical  teeth. 

(a)  Shrouding. — The  gain  in  strength  due  to  shrouding  depends 
upon  the  face  of  the  gear,  the  effect  being  more  marked  in  tha 
case  of  a  narrow  face  than  in  a  wider  one.  Wilfred  Lewis  con- 
siders shrouding  bad  practice.  However,  when  the  face  is  two 
and  one-half  times  the  circular  pitch,  he  has  demonstrated  by 
an  approximate  theoretical  investigation  that  single  shrouding 
similar  to  that  illustrated  by  Fig.  148  (a)  will  increase  the 
strength  of  the  tooth  at  least  10  per  cent,  and  double  shrouding 


(a) 


(b) 
FIG.  148. 


(c) 


as  shown  in  Fig.  148(6),  at  least  30  per  cent.  In  many  cases 
shrouding  of  gears  is  necessary,  and  the  proportions  given  in  Fig. 
148  for  the  three  methods  of  shrouding  will  serve  as  a  guide. 

(6)  Short  teeth. — Gear  teeth  whose  heights  are  less  than  those 
given  by  common  proportions  are  considerably  stronger,  and 
furthermore,  they  run  with  less  noise.  In  America,  C.  W.  Hunt 
advocates  this  type  of  tooth,  and  the  following  proportions 
for  cast  involute  teeth  are  those  he  has  successfully  used  on 
gears  for  coal-hoisting  engines  and  similar  machinery. 


Addendum  =  0.2  p' 
Face  of  gear  =  2  p'  +  1" 
Clearance  =  0.05  (pf  +  1") 


(333) 


In  Table  75  are  given  the  working,  as  well  as  maximum,  loads 
recommended  by  Mr.  Hunt  for  a  cast-iron  spur  gear  having  20 
teeth,  which  is  the  smallest  gear  he  uses.  For  proportions  of 
short  cut  teeth  recommended  by  Mr.  Hunt,  see  Table  71. 


312  METHODS  OF  STRENGTHENING  [CHAP.  XII 

TABLE  75. — STRENGTH  OF  GEAR  TEETH  USED  BY  C.  W.  HUNT 


Load  in  pounds 

Load  in  pounds 

/-«•           1 

Oi         Inr 

pitch 

Working 

Maximum 

pitch 

Working 

Maximum 

1 

1,320 

1,650 

2M 

6,700 

8,300 

IK 

2,300 

2,600 

2^ 

8,300 

10,500 

IK 

3,000 

3,700 

2% 

10,000 

12,500 

IK 

4,100 

5,000 

3 

12,000 

14,800 

(c)  Increase  of  the  angle  of  obliquity. — The  gain  in  strength  due 
to  an  increase  of  the  angle  of  obliquity  is  shown  in  Fig.  149. 
This  figure  shows  the  left  half  of  a  tooth  having  a  22j^-degree 
angle  of  obliquity  and  the  right  half  of  a  tooth  having  the  same 

pitch  but  with  an  angle  of  obliquity  equal  to  15  degrees.     Now 

2  % 

the  factor  y  which  appears  in  the  Lewis  formula  is  equal  to  ~ — > 

6p 

from  which  it  is  evident  that  an  increase  of  x,  when  the  circu- 
lar pitch  pf  remains  constant, 
in  an  increase  of  y 
I5°  and  consequently  an  increase 
in  the  strength  of  the  tooth. 
This  increase  of  x  is  shown 
in  the  figure. 

A  further  advantage  aside 
from  the  increase  of  strength 
lies  in  the  fact  that  the  size 
of  the  smallest  pinion  which 
will  mesh  with  a  rack  without 
correction  for  interference  di- 
FIG.  149.  minishes  rapidly  as  the  angle 

of  obliquity  increases.     Thus 

with  an  angle  of  obliquity  of  15  degrees,  the  30-tooth  pinion  is 
the  smallest  one  that  can  be  used  without  correction,  while  with 
an  obliquity  of  22^  degrees,  the  smallest  gear  in  an  uncorrected 
set  has  theoretically  14  teeth, 'but  practically  this  number  may 
be  reduced  to  12. 

(d)  Stub  teeth. — Another  method  of  strengthening  gear  teeth, 
which  is  now  being  used  extensively  in  automobile  transmission 
gears  and  in  gears  used  in  machine  tools  and  hoisting  machinery 
consists  of  a  combination  of  (6)  and  (c).     This  combination  gives 
what  is  known  as  the  stub  tooth.     There  are  two  systems  of  stub 


ART.  230] 


METHODS  OF  STRENGTHENING 


313 


TABLE  76. — DIMENSIONS  OF  THE 
FELLOWS  STUB  TEETH 


teeth,  differing  in  the  detail  dimensions  of  the  teeth,  as  shown 
below,  but  agreeing  on  the  choice  of  the  angle  of  obliquity, 
namely  20  degrees. 

In  one  of  these  systems,  originated  by  Mr.  C.  H.  Logue,  the 
proportions  given  in  Table  71  are 
used. 

To  the  second  system,  that 
recommended  by  the  Fellows  Gear 
Shaper  Co.,  the  tooth  dimensions 
listed  in  Table  76  apply. 

(e)  Unequal  addendum  gears. — 
Both  in  Europe  and  in  the  United 
States  certain  manufacturers  have 
advocated  the  use  of  a  system  of 
gearing  in  which  the  addendum  of 
the  driving  pinion  is  made  long, 
while  that  of  the  driven  gear  is 
made  short,  as  shown  in  Fig.  150. 
Some  of  the  advantages  claimed  for  this  system  of  gearing,  after 
several  years  of  actual  experience  with  it,  are  the  following: 

1.  This  form  of  tooth  obviates  interference,  thus  doing  away 
with  undercut  on  the  gears  having  the  smaller  numbers  of  teeth, 
and  at  the  same  time  it  increases  the  strength  of  such  gears. 


Pitch 

Thick- 
ness on 
the  pitch 
line 

Adden- 
dum 

Deden- 
dum 

« 

0.3925 

0.2000 

0.2500 

% 

0.3180 

0.1429 

0.1785 

% 

0.2617 

0.1250 

0.1562 

H 

0.2243 

0.1110 

0.1389 

*Ao 

0.1962 

0.1000 

0.1250 

Hi 

0.1744 

0.0909 

0.1137 

1%2 

0.1570 

0.0833 

0.1042 

««4 

0.1308 

0.0714 

0.0893 

(a) 


(b) 


FIG.  150. 


2.  The  sliding  friction  between  the  flanks  of  the  teeth  is  de- 
creased, since  the  arc  of  approach  is  shortened;  hence  the  wear  of 
the  teeth  is  diminished. 

3.  High-speed  gears  equipped  with  unequal  addendum  teeth 
run  more  quietly  than  standard  addendum  gears. 

4.  With  this  system  of  gearing  it  is  possible  to  make  the  teeth 


314 


METHODS  OF  STRENGTHENING 


[CHAP.  XII 


of  the  pinion  and  gear  of  equal  strength,  while  with  the  standard 
system  this  is  impossible  without  resorting  to  the  use  of  different 
materials  for  the  pinion  and  the  gear. 

The  tooth  profile  of  a  15-tooth  pinion  having  a  tooth  of  stand- 
ard proportions  is  shown  in  Fig.  150(a),  while  Fig.  150(6)  illus- 
trates the  tooth  outline  of  a  pinion  having  the  same  number  of 
teeth  and  the  same  pressure  angle,  but  with  the  addendum  and 
dedendum  based  on  the  Gleason  standard  given  in  a  following 
paragraph.  An  inspection  of  these  profiles  shows  clearly  how  the 
teeth  of  a  pinion  are  strengthened  by  means  of  this  system  of  gear 
teeth. 

From  the  above  discussion  it  is  apparent  that  the  use  of  unequal 
addendums  is  desirable  for  gears  having 
a  high  velocity  ratio.  At  the  present 
time  unequal  addendum  gears  are  used 
extensively  on  the  rear  axle  drive  of 
automobiles,  as  quite  a  number  of  manu- 
facturers have  now  adopted  this  system 
of  teeth  for  their  bevel  gears.  In 
America  up  to  the  present  time,  the  un- 
equal addendum  teeth  are  used  chiefly 
with  bevel  gears,  but  there  is  no  reason 
why  they  should  not  be  used  to  advan- 
tage in  certain  spur  gear  drives  on  ma- 
chine tools  and  in  other  classes  of  ma- 
chinery. As  yet  very  little  progress  has  been  made  in  this 
direction. 

Gleason  standard. — The  Gleason  Works  have  adopted  as  their 
standard  for  high  ratio  bevel  gears  the  following  proportions  for 
unequal  addendum  teeth. 


TABLE  77. — CONSTANTS 
FOR  DETERMINING  TOOTH 
THICKNESS  FOR  GLEASON 
UNEQUAL  ADDENDUM 
TEETH 


Angle  of 
tooth 
thrust 

Constant  for 

Pinion 

Gear 

14^° 
15° 
20° 

0.5659 
0.5683 
0.5927 

0.4341 
0.4317 
0.4073 

Addendum  for  pinion  =  0.7  working  depth 
Addendum    for  gear  =  0.3  working  depth 


(334) 


The  working  depth  is  assumed  to  be  twice  the  reciprocal  of  the 
diametral  pitch,  or  the  circular  pitch  multiplied  by  the  factor 
0.3183. 

To  determine  the  thickness  of  the  tooth  on  the  pitch  circle, 
when  these  formulas  are  used,  multiply  the  circular  pitch  by  the 
constants  given  in  Table  77. 

(/)  Buttressed  tooth. — The  buttress  or  hook-tooth  gear  can  be 
used  in  cases  where  the  power  is  always  transmitted  in  the  same 


ART.  231] 


SPECIAL  GEARS 


315 


direction.  The  load  side  of  the  tooth  has  the  usual  standard 
profile,  while  the  back  side  of  the  tooth  has  a  greater  angle  of 
obliquity  as  shown  in  Fig.  151.  To  compare  its  strength  with 
that  of  the  standard  tooth,  use  the  following  method:  Make  a 
drawing  of  the  two  teeth  and  measure  their  thicknesses  at  the 
tops  of  the  fillets;  then  the  strength  of  the  hook  tooth  is  to  the 
standard  as  the  square  of  the  tooth  thickness  is  to  the  square  of 
the  thickness  of  the  standard  tooth. 

(g)  Helical  teeth. — Properly  supported  gears  having  accurately 
made  helical  teeth  will  run  much  smoother  than  ordinary  spur 
gears.  In  the  latter  form  of  gearing  there  is  a  time  in  each  period 
of  contact  when  the  load  is  concentrated  on  the  upper  edge  of 
the  tooth,  thus  having  a  leverage  equal  to  the  height  of  the  tooth. 


Involute 


15°  Involute^ 


FIG.  151. 

With  helical  gearing,  however,  the  points  of  contact  at  any  in- 
stant are  distributed  over  the  entire  working  surface  of  the  tooth 
or  such  parts  of  two  teeth  in  contact  at  the  same  time.  There- 
fore, the  mean  lever  arm  with  which  the  load  may  act  in  order  to 
break  the  tooth  cannot  be  more  than  half  the  height  of  the  tooth. 
It  follows  that  the  helical  teeth  are  considerably  stronger  than  the 
straight  ones.  The  subject  of  helical  gears  will  be  discussed  more 
in  detail  in  Chapter  XIV. 

231.  Special  Gears. — Specially  designed  gears,  differing  radi- 
cally from  those  discussed  in  the  preceding  articles,  are  used  when 
it  is  desired  to  provide  some  slippage  so  as  to  protect  a  motor 
against  excessive  overload,  or  to  prevent  breakage  of  some  part 
of  the  machine.  Special  gears  are  also  required  where  heavy 
shocks  must  be  absorbed,  thus  again  protecting  the  machine 
against  possible  damage.  The  first  type  of  gear  mentioned  is 


316 


SLIP  GEARS 


[CHAP.  XII 


known  as  a  slip  gear  and  the  second,  as  a  flexible  or  spring  cush- 
ioned gear. 

(a)  Slip  gears. — A  slip  gear  is  a  combination  of  a  gear  and  a 
friction  clutch,  the  latter  being  so  arranged  that  it  is  always  in 
engagement,  but  will  slip  when  an  extra  heavy  load  comes  upon 
the  gear.  Slip  gears  are  used  to  some  extent  in  connection  with 
electric  motor  drives,  and  in  such  installations  they  really  serve 


FIG.  152. 

as  safety  devices  by  protecting  the  motor  from  dangerous  over- 
loads. Two  rather  simple  designs  of  slip  gears  are  illustrated  in 
Figs.  152  and  153. 

1.  Pawling s-Harnischfeger  type. — The  design  shown  in  Fig.  152 
is  used  by  the  Pawlings-Harnischfeger  Co.  in  connection  with 
some  of  their  motor-driven  jib  cranes.  The  gear  a,  meshing 
directly  with  the  motor  pinion,  is  mounted  upon  the  flanged  hub 
b  and  the  bronze  cone  c,  both  of  which  are  keyed  to  the  driven 
shaft  d  as  shown  in  the  figure.  By  means  of  the  three  tempered 


ART.  231] 


SLIP  GEARS 


317 


steel  spring  washers  e  and  the  two  adjusting  nuts  /,  the  desired 
axial  force  may  be  placed  on  the  clutch  members  b  and  c.  In 
reality  the  combination  a,  b}  and  c  is  nothing  more  than  a  com- 
bined cone  and  disc  clutch,  the  analysis  of  which  is  given  in  de- 
tail in  Chapter  XVI.  The  angle  that  an  element  of  the  cone 
makes  with  the  axis  is  15  degrees  for  the  design  shown  in  Fig.  152. 
2.  Ingersoll  type. — A  second  design  of  slip  gear  differing  slightly 
from  the  above  is  shown  in  Fig.  153.  It  is  used  by  the  Ingersoll 
Milling  Machine  Co.  on  the  table  feed  mechanism  of  their 
heavy  milling  machines.  Its  function  is  to  permit  the  pinion  d 
to  slip  when  the  load  on  the  cutter  becomes  excessive.  The 


FIG.  153. 

driving  shaft  a  has  keyed  to  it  a  bronze  sleeve  6  upon  which  slides 
the  sleeve  c,  also  made  of  bronze.  As  shown,  a  part  of  the  length 
of  the  sleeves  b  and  c  is  turned  conical  so  as  to  fit  the  conical  bore 
of  the  steel  pinion  d.  The  frictional  force  necessary  to  operate 
the  table  is  obtained  by  virtue  of  the  pressure  of  the  spring  e 
located  on  the  inside  of  the  sleeve  c.  By  means  of  the  adjusting 
nuts  /  and  g,  the  spring  pressure  may  be  varied  to  suit  any  condi- 
tion of  operation. 

(b)  Flexible  gears. — The  so-called  flexible  or  spring-cushioned 
gear  is  used  on  heavy  electric  locomotives,  and  its  chief  function 
is  to  relieve  the  motor  and  entire  equipment  from  the  enormous 
shocks  due  to  suddenly  applied  loads.  In  Fig.  154  is  shown  a 
well-designed  gear  of  this  kind  as  made  by  The  R.  D.  Nuttall  Co. 
The  gear  consists  of  a  forged-steel  rim  a  on  the  inner  surface  of 
which  are  a  number  of  short  arms  or  lugs  6  as  illustrated  in  Fig. 


318 


FLEXIBLE  GEARS 


[CHAP.  XII 


154(6),  which  represents  a  section  through  the  gear  along  the  line 
OB.  The  gear  rim  a  is  mounted  upon  the  steel  casting  hub  c 
which  is  equipped  with  projecting  arms  d.  These  arms  are 
double,  as  shown  in  the  section  through  OB,  and  are  provided 
with  sufficient  clearance  to  accommodate  the  projecting  lugs  b. 
The  heavy  springs  e  form  the  only  connection  between  the  double 
arms  d  and  the  lugs  6;  hence,  all  of  the  power  transmitted  from 
the  rim  to  the  hub  must  pass  through  the  springs  e.  Special 
trunnions  or  end  pieces  are  used  on  the  springs  to  give  a  proper 
bearing  on  the  lugs  and  arms.  The  cover  plate  /  bolted  to  the 
hub  c  affords  a  protection  to  the  interior  of  the  gear  against 
dust  and  grit.  It  is  evident  from  this  description  that  the  springs 


FIG.  154. 

provide  the  necessary  cushioning  effect  required  to  absorb  the 
shocks  caused  by  suddenly  applied  overloads.  The  remarks 
relating  to  the  design  of  spring-cushioned  sprockets,  as  given  in 
Art.  193,  also  apply  in  a  general  way  to  the  design  of  flexible 
gears. 

EFFICIENCY  OF  SPUR  GEARING 

There  is  probably  no  method  of  transmitting  power  between 
two  parallel  shafts  that  shows  a  better  efficiency  than  a  pair  of 
well-designed  and  accurately  cut  gears.  So  far  as  the  author 
knows,  no  extensive  investigation  has  ever  been  made  of  the  effi- 
ciency of  spur,  bevel,  and  helical  gearing;  at  least,  very  little  in- 
formation has  appeared  in  the  technical  press  on  this  important 
subject.  It  is  generally  assumed  that  the  efficiency  of  gearing 
becomes  less  as  the  gear  ratio  increases,  and  the  correctness  of 
this  assumption  is  proved  by  a  mathematical  analysis  proposed 
by  Weisbach, 


ART.  232] 


EFFICIENCY  OF  GEARS 


319 


232.  Efficiency  of  Spur  Gears. — By  means  of  the  analysis 
following,  it  is  possible  to  arrive  at  the  expression  for  the  amount 
of  work  lost  due  to  the  friction  between  the  teeth.  Knowing 
this  lost  work,  also  the  useful  work  transmitted  by  the  gears,  we 
have  a  means  of  arriving  at  the  probable  efficiency  of  a  pair  of 
gears. 

In  Fig.  155  are  shown  two  spur  gears  1  and  2  transmitting 
power.  In  this  figure  the  line  MN,  making  an  angle  0  with  the 
common  tangent  CT,  represents  the  line  of  action  of  the  tooth 
thrust  between  the  gears. 


FIG.  155. 

Let  HI  and  HZ  denote  the  revolutions  per  second  of  the  gears. 
TI  and  T2  denote  the  number  of  teeth  in  the  gears  1  and  2, 

respectively. 

0)1  and  o)2  denote  the  angular  velocity  of  the  gears  1  and  2, 
respectively. 

p'  =  the  circular  pitch, 
s  =  the  distance  from  the  pitch  point  C  to  the  point 

of  contact  of  two  teeth. 
fj.  =  coefficient  of  sliding  friction. 

To  find  the  velocity  of  sliding  at  the  point  of  contact  of  two 
teeth,  we  employ  the  principle  that  the  relative  angular  velocity 
of  the  gears  1  and  2  is  equal  to  the  sum  or  difference  of  the  angular 
velocities  o>i  and  o)2  of  the  wheels  relative  to  their  fixed  centers 
Oi  and  02;  thus  if  o>  denotes  this  relative  angular  velocity, 

0)    ==    O)i     ~f"    O)2« 


320  EFFICIENCY  OF  GEARS  [CHAP.  XII 

the  minus  sign  being  used  when  one  of  the  wheels  is  annular  and 
coi  and  co2  have  the  same  sense.  The  velocity  v'  with  which  one 
tooth  slides  on  the  other  is  then  the  product  of  this  angular  veloc- 
ity co  and  the  distance  s  between  the  point  of  contact  and  the 
pitch  point,  which  is  the  instantaneous  center  of  the  relative 
motion  of  1  and  2;  that  is, 

v'  =  8  (o>i  ±  co2)  (335) 

The  distance  s  varies  ;  at  the  pitch  point  it  is  zero,  and  when  the 
teeth  quit  contact  it  has  a  value  of  0.7  to  0.9  p'  with  teeth  having 
the  ordinary  proportions.  The  average  value  of  s  may  be  taken 
as  0.4  p'. 

Since  P  is  the  normal  pressure  between  the  tooth  surfaces,  the 
force  of  friction  is  /iP,  and  the  work  of  friction  per  second  is 

Wt  =  uPv'  (336) 

The  formula  for  Wt  may  be  put  into  more  convenient  form  by 
combining  (335)  and  (336),  and  substituting  in  the  resulting  equa- 
tion the  following  values  of  coi,  co2  and  n2  : 

m 

o>i  =  2irni'  co2  =  27rn2;  n2  =  HI  -=- 

1  2 

Hence, 

(337) 


in  which  v  represents  the  velocity  of  a  point  on  the  pitch  line. 

The  component  of  P,  in  the  direction  of  the  common  tangent 
CT  to  the  pitch  circles  of  the  gears,  is  Pcos/3;  hence  the  work  per 
second  that  this  force  can  do  is 

W0  =  PvcosQ  (338) 

Adding  (337)  and  (338),  it  is  evident  that  the  work  Wr  put  into 
the  gears,  omitting  the  friction  on  the  gear  shafts,  is 

W.'  =  W0  +  Wt  (339) 

The  component  of  P  in  a  radial  direction  is  Psin/3.  The  total 
pressure  upon  the  bearings  of  each  shaft  is  P;  hence  the  work 
lost  in  overcoming  the  frictional  resistances  of  these  bearings  is 
as  follows: 

TF6=irM/JcWi+  M»),  (340) 

in  which  di  and  dz  represent  the  diameters  of  the  shafts,  and  ju' 
the  coefficient  of  journal  friction. 


ART.  232]  EFFICIENCY  OF  GEARS  321 

With  friction  considered,  it  follows  that  the  total  work  required 
to  transmit  the  useful  work  WQ  is 

W  =  WQ  +  Wt  +  Wb  (341) 

The  efficiency  of  the  pair  of  gears  including  the  bearings  is 
therefore 

'  =  w  (342) 

If  it  is  desirable  to  estimate  the  efficiency  of  the  gears  exclusive 
of  the  bearings,  the  following  expression  may  be  used  : 


2.51  UL  sec 


(343) 


For  gears  having  cast  teeth,  the  coefficient  of  friction  ju  may 
vary  from  0.10  to  0.20,  while  for  cut  gears  the  value  may  be  less 
than  one-half  those  just  given.  However,  even  with  the  larger 
coefficient  of  friction  quoted,  the  loss  due  to  friction  is  small. 
It  is  apparent  from  (343)  that  the  efficiency  is  increased  by  em- 
ploying gears  having  relatively  largS  numbers  of  teeth. 

References 

American  Machinist  Gear  Book,  by  C.  H.  LOGUE. 

A  Treatise  on  Gear  Wheels,  by  G.  B.  GRANT. 

Machine  Design,  by  SMITH  and  MARX. 

Spur  and  Bevel  Gearing,  by  Machinery. 

Elements  of  Machine  Design,  by  J.  F.  KLEIN. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Elektrisher  Antriebmittels  Zahnradubertragung,  Zeit.  des  Ver.  deutsch 
Ing.,  p.  1417,  1899. 

Interchangeable  Involute  Gear  Tooth  System,  A.  S.  M.  E.,  vol.  30,  p.  921. 

Interchangeable  Involute  Gearing,  A.  S.  M.  E.,  vol.  32,  p.  823. 

Proposed  Standard  Systems  of  Gear  Teeth,  Amer.  Mach.,  Feb.  25,  1909. 

Tooth  Gearing,  A.  S.  M.  E.,  vol.  32,  p.  807. 

Gears  for  Machine  Tool  Drives,  A.  S.  M.  E.,  vol.  35,  p.  785. 

The  Strength  of  Gear  Teeth,  A.  S.  M.  E.,  vol.  34,  p.  1323. 

The  Strength  of  Gear  Teeth,  A.  S.  M.  E.,  vol.  37,  p.  503. 

Recent  Developments  in  the  Heat  Treatment  of  Railway  Gearing,  Proc. 
The  Engrs.  Soc.  of  W.  Pa.,  vol.  30,  p.  737. 

Gear  Teeth  Without  Interference  or  Undercutting,  Mchy.,  vol.  22,  p.  391. 

Spur  Gearing,  Trans.  Inst.  of  Mech.  Engr.,  May,  1916. 

Safe  and  Noiseless  Operation  of  Cut  Gears,  Amer.  Mach.,  vol.  45,  p.  1029. 

Efficiency  of  Gears,  Amer.  Mach.,  Jan.  12,  1905;  Aug.  19,  1909. 

Internal  Spur  Gearing,  Mchy.,  vol.  23,  p.  405. 

Chart  for  Selecting  Rawhide  Pinions,  Mchy.,  vol.  23,  p.  223. 


CHAPTER  XIII 
BEVEL   GEARING 

When  two  shafts  which  intersect  each  other  are  to  be  connected 
by  gearing,  the  result  is  a  pair  of  bevel  gears.  Occasionally, 
however,  the  shafts  are  inclined  at  an  angle  to  each  other  but  do 
not  intersect,  in  which  case  the  gears  are  called  skew  bevels. 
The  form  of  tooth  which  is  almost  universally  used  for  bevel 
gears  is  the  well-known  involute.  This  is  probably  due  to  the 
fact  that  slight  errors  in  its  form  are  not  nearly  so  detrimental  to 
the  proper  running  of  the  gears  as  when  the  tooth  curves  are 
cycloidal. 

233.  Methods  of  Manufacture. — Bevel  gears  may  be  either 
cast  or  cut.  The  process  of  casting  is  not  materially  different 
from  that  used  in  spur  gearing,  but  the  process  of  cutting  is  much 
more  difficult  on  account  of  the  continuously  changing  form  and 
size  of  the  tooth  from  one  end  to  the  other. 

As  in  the  case  of  spur  gearing,  there  are  several  different 
methods  of  cutting  the  teeth,  some  of  which  form  the  teeth  with 
theoretical  accuracy,  while  others  produce  only  approximately 
correct  forms.  Three  of  the  methods  give  very  accurate  results, 
but  they  require  expensive  special  machines  and  are  used  only 
when  very  high-grade  work  is  desired.  The  three  methods  are: 
the  templet-planing  process,  represented  by  the  Gleason  gear 
planer;  the  templet-grinding  process,  now  used  but  little,  repre- 
sented by  a  machine  manufactured  by  the  Leland  and  Faulconer 
Co.;  and  the  moulding-planing  process,  represented  by  the  Bil- 
gram  bevel  gear  planer. 

In  each  of  these  processes  the  path  of  the  cutting  tool  passes 
through  the  apex  of  the  cone,  that  is,  the  point  of  intersection  of 
the  two  shafts,  and  consequently  the  proper  convergence  is 
given  to  the  tooth.  With  a  formed  rotating  cutter,  it  is  impossi- 
ble to  produce  the  proper  convergence  and  in  many  cases  the 
teeth  have  to  be  filed  after  they  are  cut,  before  they  will  mesh 
properly.  Nevertheless,  the  milling  machine  is  very  commonly 
used  for  cutting  bevel  gears,  for  the  simple  reason  that  the 

322 


ART.  234] 


BEVEL  GEAR  TEETH 


323 


equipment  of  most  shops  includes  a  milling  machine,  while 
comparatively  few  shops  do  enough  bevel  gear  cutting  to  justify 
the  purchase  of  an  expensive  special  machine  for  that  purpose. 

234.  Form  of  Teeth. — When  the  gears  are  plain  bevel  frictions, 
it  is  evident  that  the  faces  of  the  gears  must  be  frustums  of  a 
pair  of  cones  whose  vertices  are  at  the  point  of  intersection  of  the 
axes.  These  cones  may  now  be  considered  the  pitch  cones  of  a 
pair  of  tooth  gears,  and  the  teeth  may  be  generated  in  a  manner 
analagous  to  the  methods  used  for  spur  gearing.  In  discussing 


r— iE_  _  G    _\J 


the  method  of  forming  the  teeth,  the  involute  system  only  will  be 
considered,  since  the  cycloidal  forms  are  seldom  used. 

In  Fig.  156,  let  the  cone  OH  I  represent  the  so-called  base  cone 
of  the  bevel  gear  shown,  from  which  the  involute  tooth  surfaces 
are  to  be  developed.  In  order  to  simplify  the  conception  of  the 
process  of  developing,  imagine  the  base  cone  to  be  enclosed  in  a 
very  thin  flexible  covering  which  is  cut  along  the  line  OE.  Now 
unwrap  the  covering,  taking  care  to  keep  it  perfectly  tight;  then 
the  surface  generated  by  the  edge  or  element  OF  is  the  desired 
involute  surface.  The  point  E,  while  it  evidently  generates  an 
involute  of  the  circle  HI,  is  also  constrained  to  remain  a  constant 


324 


BEVEL  GEAR  TEETH 


[CHAP.  XIII 


distance  from  0  equal  to  OE,  or  in  other  words,  it  travels  on  the 
surface  of  a  sphere  HAL  For  that  reason  the  curve  EF  is  called 
a  spherical  involute.  The  spherical  surface  which  should  theo- 
retically form  the  tooth  profile  is  a  difficult  surface  to  deal  with  in 
practice  on  account  of  its  undevelopable  character,  and  as  is 
shown  in  the  figure  no  appreciable  error  is  introduced  if  the  con- 
ical surface  CBD  is  substituted  for  the  spherical  surface  CAD. 
The  cone  CBD,  which  is  called  the  back  cone,  is  tangent  to  the 


sphere  at  the  circle  CD,  and  the  pitch  distance  practically  coin- 
cides with  the  sphere  for  the  short  distance  necessary  to  include 
the  entire  tooth  profile.  When  it  is  desired  to  obtain  the  form  of 
the  teeth,  as  is  necessary  in  case  a  wood  pattern  or  a  formed  cutter 
is  to  be  made,  the  back  cone  is  developed  on  a  plane  surface  as 
shown  in  Fig.  157.  .  It  is  evident  that  the  surface  which  contains 
the  tooth  profile  has  a  radius  of  curvature  equal  to  BD,  so  the 
profile  must  be  laid  off  on  a  circle  of  that  radius  in  precisely  the 
same  manner  as  that  used  for  spur  gearing.  However,  this  pro- 
file is  correct  for  one  point  only,  namely,  at  the  large  end.  In 


ART.  235]  DEFINITIONS  325 

order  to  determine  the  form  of  the  tooth  for  its  entire  length,  it  is 
necessary  to  have  the  profile  of  the  tooth  at  each  end.  This 
may  be  obtained  by  developing  the  back  cone  AOG  and  proceed- 
ing as  before.  The  two  profiles  just  discussed  are  laid  out  from 
the  line  LI  as  shown.  The  back  cone  radius  LK  is  equal  in  length 
to  AG,  and  LJ  is  equal  to  BD.  If  a  wood  pattern  is  to  be  made, 
templets  are  formed  of  the  exact  profile  of  the  tooth  at  the  large 
and  small  ends.  These  templets  are  then  wrapped  around  the 
gear  blank  and  the  material  is  cut  out  to  the  shape  of  the  templets. 

235.  Definitions. — (a)  By  the  expression  back  cone  radius  is 
meant  the  length  of  an  element  of  the  back  cone,  as  for  example 
the  line  //  in  Fig.  157. 

(b)  The  edge  angle  is  the  angle  between  a  plane  which  is  tangent 
to  the  back  cone  and  the  plane  containing  the  pitch  circle.     In 
Fig.  157  this  angle  is  designated  by  the  symbols  0i  and  02  for 
the  pinion  and  gear*  respectively. 

(c)  The  center  angle  is  the  angle  between  a  plane  tangent  to  the 
pitch  cone  and  the  axis  of  the  gear.     For  the  pinion  and  gear 
shown  in  Fig.  157,  the  center  angle  is  designated  as  «i  and  «2, 
respectively.     From  the  geometry  of  the  figure  it  is  evident  that 
61  =  a  i,  and  02  =  «2. 

(d)  The  cutting  angle,  represented  by  the  symbols  Xi  and  X2  in 
Fig.  157,  is  the  angle  between  a  plane  tangent  to  the  root  cone  and 
the  axis  of  the  gear. 

(e)  By  the  term  face  angle  is  meant  the  angle  between  the  plane 
containing  the  pitch  circle  and  the  outside  edge  of  the  tooth,  as 
represented  by  the  symbols  <pi  and  <p2  in  Fig.  157. 

(/)  Backing  is  the  distance  from  the  addendum  at  the  large 
end  of  the  teeth  to  the  end  of  the  hub,  as  represented  by  the 
dimensions  LI  and  L2  in  Fig.  157. 

(g)  The  expression  formative  number  of  teeth  is  the  number  of 
teeth  of  the  given  pitch  which  would  be  contained  in  a  complete 
spur  gear  having  a  radius  equal  to  the  back  cone  radius.  This 
number  of  teeth  is  used  in  selecting  the  proper  cutter  for  cutting 
the  gear  and  also  for  obtaining  the  value  of  the  Lewis  factor  when 
calculating  the  strength  of  the  bevel  gear. 

BEVEL-GEAR  FORMULAS 

The  following  formulas,  expressing  the  relations  existing  be- 
tween the  various  dimensions  and  angles  of  bevel  gears,  are  im- 


326 


ACUTE-ANGLE  BEVEL  GEARS 


[CHAP.  XIII 


portant  and  are  necessary  for  determining  the  complete  dimen- 
sions required  to  manufacture  such  gears.  In  arriving  at  these 
formulas,  two  general  types  of  bevel  gears  must  be  considered: 

1.  That  type  in  which  the  angle  between  the  intersecting  shafts 
is  less  than  90  degrees,  as  shown  in  Fig.  158. 

2.  That  type  having  an  angle  between  the  shafts  greater  than 
90  degrees,  an  illustration  of  which  is  shown  in  Fig.  159. 

Having  obtained  the  formulas  for  either  of  these  types  of  gears, 
those  for  the  more  common  case,  namely  when  the  shafts  make  an 
angle  of  90  degrees,  may  readily  be  derived. 

236.  Acute-angle  Bevel  Gears. — In  Fig.  158  is  shown  a  pair  of 
bevel  gears  in  which  the  angle  6  between  the  shafts  is  less  than 
90  degrees.  In  deriving  the  desired  relations,  the  following  no- 
tation will  be  used: 


FIG.  158. 

D  =  the  pitch  diameter. 
D'  =  the  outside  diameter. 
T  =  the  number  of  teeth. 
e  =  the  diameter  increment. 
c  =  the  clearance  at  the  top  of  the  tooth, 
the  diametral  pitch, 
the  circular  pitch, 
the  addendum. 


P 

P' 


ART.  236]  ACUTE-ANGLE  BEVEL  GEARS  327 

In  this  discussion  the  subscripts  1  and  2,  when  applied  to  the 
various  symbols,  refer  to  the  pinion  and  gear,  respectively.  From 
the  geometry  of  the  figure,  we  obtain  the  following  relations : 

a  = 


2  sin  8 

2  tan  d 

Di  Dl  sin  0 


tan«i  = 

x  (a  -\-  0)       JJ2  -f-  Ui  cos  u 

*""—  (344) 


fj  +  cos  0 

The  equation  just  established  enables  us  to  determine  the  mag- 
nitude of  the  center  angle  of  the  pinion.  Subtracting  a\  from  the 
angle  0  included  between  the  two  shafts  gives  the  magnitude  of 
the  center  angle  «2  of  the  gear. 

If  it  is  desired  to  determine  the  angle  #2  by  means  of  calcula- 
tions, the  following  formula,  derived  in  the  same  manner  as  (344), 
may  be  used  : 

^  (345) 


jT  +  COS  8 

Determining  the  magnitudes  of  «i  and  a2  by  means  of  (344)  and 
(345),  the  calculations  may  be  checked  very  readily,  since 

Oil   +    «2    =    8. 

To  determine  the  angle  0i  of  the  pinion,  we  must  find  the 
angle  increment,  by  which  is  meant  the  angle  included  between 
the  pitch  cone  element  and  the  face  of  the  tooth.  Thus 

2  s 

tan  (fr  -  «i)  =  -p-  sin  a,,  (346) 

from  which  the  angle  increment  may  be  obtained.  The  addition 
of  (0i  —  «i)  to  the  center  angle  gives  the  magnitude  of  the 
angle  0i. 

The  angle  decrement  (on  —  Xi)  may  be  determined  from  the 
following  relation: 


tan  (ttl  -  Xi)  =        n        sin  ai  (347) 

u\ 

By  subtracting  (a.\  —  Xi)  from  the  center  angle  the  magnitude 
of  the  cutting  angle  Xi  is  found. 


328  OBTUSE-ANGLE  BEVEL  GEARS  [CHAP.  XIII 

Since  the  angle  increment  and  angle  decrement  of  the  pinion 
are  exactly  the  same  as  the  corresponding  angles  of  the  gear,  the 
face  and  cutting  angles  of  the  latter  may  be  found. 

In  turning  the  blanks,  it  is  necessary  that  the  outside  diameter 
of  both  the  pinion  and  the  gear  be  known.  These  diameters  are 
obtained  by  adding  twice  the  diameter  increment  to  the  pitch 
diameters.  The  diameter  increment  is  calculated  by  the  follow- 
ing equations: 

For  the  pinion,  e\  =  scosa:i 


For   the    gear,  e^  =  ' 


From  these  relations  we  get 

D1  =  DI  +  2  ov,v,o^i  ,  ("UQ^ 

£>;=D2  +  2scosa2/ 

The  length  of  the  face  of  the  pinion  measured  parallel  to  the 
axis  is  FcosfSi  and  the  corresponding  dimension  for  the  gear  is 
Fcosfo. 

237.  Obtuse -angle  Bevel  Gears. — By  the  expression  obtuse- 
angle  bevel  gearing  is  meant  a  gear  and  pinion  in  which  the  angle 
between  the  shafts  is  more  than  90  degrees.  It  is  evident  from 
this  that  the  following  three  forms  of  such  gearing  are  possible: 

(a)  In  the  first  form,  which  is  more  common  than  either  of  the 
other  two,  the  center  angle  a2  of  the  gear  is  made  less  than  90 
degrees.  For  convenience  of  reference,  we  shall  call  this  form 
the  regular  obtuse-angle  bevel  gear. 

(6)  In  the  second  form,  which  is  rarely  used,  the  center  angle 
0:2  of  the  gear  is  made  90  degrees.  In  this  case  the  pitch  cone 
becomes  a  plain  disc;  such  a  gear  is  then  known  as  a  crown  gear. 

(c)  In  the  third  form,  which  should  be  avoided  whenever  pos- 
sible, the  center  angle  «2  of  the  gear  is  greater  than  90  degrees. 
In  such  a  gear  the  teeth  must  be  formed  on  the  internal  conical 
surface,  thus  giving  it  the  name  of  internal  bevel  gear.  An  internal 
bevel  gear  can  generally  be  avoided  without  changing  the  posi- 
tions of  the  shafts  by  using  an  acute-angle  gear  set,  in  which  the 
angle  between  the  shafts  is  made  equal  to  the  supplement  of  the 
original  angle  between  the  shafts. 

Using  the  same  notation  as  in  the  preceding  article,  the  im- 
portant formulas  for  the  bevel  gears  illustrated  in  Fig.  159,  in 
which  the  angle  6  is  greater  than  90  degrees,  are  as  follows : 


ART.  237]  OBTUSE-ANGLE  BEVEL  GEARS 

For  the  pinion 


329 


tan  ai  = 


sin  (180  -  61) 


~  -  cos  (180  -  0) 


1  1 


sin  6 


cos 


(350) 


Generally  speaking,  the  first  form  of  equation  (350)  is  preferred 
by  most  designers  and  shop  men,  although  the  second  form,  which 
is  the  same  as  (344),  is  really  more  convenient.  In  the  solution 
of  any  problem  pertaining  to  obtuse-angle  bevel  gearing,  it  is 
well  to-  determine  what  form  of  obtuse  bevel  gear  is  being  ob- 


FIG.  159. 

tained  before  proceeding  with  the  calculations,  as  forms  (b)  and 
(c)  discussed  above  require  special  formulas.  To  find  out  what 
form  of  gear  is  being  obtained  proceed  in  the  following  manner: 

To  the  magnitude  of  «i,  obtained  from  (350),  add  90  degrees 
and  if  the  sum  thus  obtained  is  in  excess  of  the  given  angle  0, 
then  the  resulting  gears  will  be  of  form  (a),  namely  ordinary  ob- 
tuse bevel  gears.  If,  however,  the  sum  «i  +  90  is  equal  to  the 
given  angle  0,  the  result  will  be  a  crown  gear  and  pinion.  An 
internal  bevel  gear  will  result  when  (on  +  90)  <0. 

For  the  ordinary  obtuse-angle  bevel  gear,  the  center  angle  az 


330  STRENGTH  OF  BEVEL  GEARS  [CHAP.  XIII 

of  the  gear,  if  desired,  may  be  determined  by  means  of  the  fol- 
lowing formula: 

sin  (180  -  6)  sin0 

tan«2  =  7p~  =  TjT-  (351) 

^-  cos  (180-  0)       ^  +  cosd 

1  2  1  2 

The  remaining  calculations  for  the  ordinary  obtuse-angle  gears 
are  made  by  means  of  the  formulas  given  in  the  preceding  article. 

238.  Right-angle  Bevel  Gears.— The  great  majority  of  the 
bevel  gears  in  common  use  in  machine  construction  have  their 
shafts  at  right  angles  to  each  other  as  shown  in  Fig.  157.  The 
formulas  in  this  case  may  be  derived  directly  from  those  in  Art. 
236,  by  substituting  for  6  its  magnitude  90  degrees;  hence  (344) 
and  (345)  reduce  to  the  following  simple  forms: 


Tl 

tan  ai  =  7=- 

1  2 

T 

tan  «2  = 


(352) 


The  remaining  formulas  given  in  Art.  236  will  apply  to  the 
present  case  without  change  or  modification. 

STRENGTH  OF  BEVEL  GEARING 

239.  General  Assumptions. — As  in  the  case  of  spur  gearing, 
formulas  for  the  strength  of  bevel-gear  teeth  will  be  derived  for 
the  following  two  cases:  (a)  When  the  teeth  are  cast;  (6)  when 
the  teeth  are  cut.  In  analyzing  the  strength  of  both  kinds  of 
teeth,  we  shall  assume  that  the  gear  is  supported  rigidly  and  that 
the  load  coming  upon  it  will  not  distort  the  teeth.  Distortion 
of  the  teeth  means  that  the  elements  of  the  tooth  form  will  no 
longer  intersect  at  the  apex  of  the  pitch  cone.  The  above  as- 
sumption also  means  that  the  distribution  of  the  load  on  the  tooth 
produces  equal  stresses  at  all  points  along  the  line  of  the  weakest 
section.  The  last  statement  may  be  proved  by  the  following 
analysis:  From  Fig.  160  or  162  it  is  evident  that  the  dimensions 
of  the  cross-section  of  the  tooth,  at  any  section,  are  proportional 
to  the  distance  that  the  section  is  from  the  apex  0;  hence  we  ob- 
tain the  following  series  of  equations: 

t  =  ~- 

(353) 


ART.  240] 


STRENGTH  OF  BEVEL  GEARS 


331 


Furthermore,  the  deflection  A  of  the  tooth  at  the  point  where  the 
line  of  action  of  the  force  dW  intersects  the  center  line  of  the 
tooth  is  also  proportional  to  the  distance  I. 

The  deflection  of  the  small  section  dl  is  given  by  the  expression 

h*dW 


A  = 


3  El 


=  kl 


(354) 


Substituting  in  (354)  the  value  of  h  from  (353)  and  the  value  of 
/  in  terms  of  the  dimensions  of  the  section,  it  follows  that 


(355) 


FIG.   160. 


Applying  the  formula  for  flexure  to  the  elementary  cantilever 
beam,  we  obtain 


hdW 


6 


Combining  (353)  and  (356),  we  find 
dW  _    jtiH 

^T  "  eT^i 


=  CIS 


Comparing  (355)  and  (357),  it  follows  that 

T7-J 

S  =  -     =  constant 


(356) 


(357) 


(358) 


240.  Strength  of  Cast  Teeth. — It  is  sufficiently  accurate  to 
consider  the  cast  bevel  gear  tooth  as  a  cantilever  beam,  the  cross- 


332  STRENGTH  OF  CAST  TEETH  [CHAP.  XIII 

sections  of  which  are  rectangular  and  converge  toward  the  apex 
of  the  pitch  cone.  Furthermore,  the  load  to  be  transmitted  is 
assumed  as  acting  tangentially  at  the  tip  of  the  tooth.  The 
formula  for  the  strength  of  cast  teeth  based  upon  the  above  as- 
sumption, as  well  as  that  given  in  the  preceding  article,  may  be 
derived  as  follows: 

By  equating  the  bending  moment  on  a  small  element  dl  of  the 
tooth  to  its  moment  of  resistance  and  solving  for  the  elementary 
force  dW,  we  have  from  (356)  that 


Also,  from  (357)  we  get 


Now  the  moment  of  the  elementary  force  dW  about  the  apex 
0  is  IdW]  hence  the  elementary  moment 


dM  =  TO 

Integrating  this  expression  between  the  limits  h  and  12,  we  ob- 
tain 

0,2 

M  =  -~(ll-  ID  (360) 


Since  M  represents  the  total  turning  moment  about  the  apex 
0  of  the  pitch  cone,  we  may  readily  determine  the  magnitude  of 
the  force  acting  at  any  point,  as  for  example  at  the  large  diameter 
of  the  gear,  by  merely  dividing  M  by  the  distance  from  that  point 
to  the  apex.  Let  W\  denote  the  force  which,  if  applied  at  the 
large  end  of  the  tooth,  will  produce  a  turning  moment  equal  to 
M  ;  then 


Qj2         /3  78 

=  ^  ^-j 
iBhii    i\ 


Substituting  in  (361)  the  value  of  12  =  l\  —  f,  and  simplifying 
the  resulting  equation, 

The  proportions  of  cast  bevel  gear  teeth  are  the  same  as  those 
given  for  cast  spur  gears  in  Art.  222,  namely  hi  =  0.7  p'  and 


ART.  240]  STRENGTH  OF  CAST  TEETH  333 

ti  =  0.475  p'.     Substituting  these  values  in  (362),  we  get 

TFi  =  0.018  Sp'f  [3  -  •  —  +  y  (363) 

Letting  m  denote  the  quantity  0.018  [3  -   y  +  ~j\,  (363)  re- 
duces to  the  following  form: 

Wi  =  Sp'fm  (364) 

A  study  of  the  prevailing  practice  among  manufacturers  of 
cast  bevel  gears  shows  that  the  face  of  Such  gears  is  made  from 


0.05- 


0.04 


•0.03- 


0.2 

Ratio    of 
FIG.  161. 


0.3 


0.4 


f  to  ii 


two  to  three  times  the  circular  pitch,  depending  upon  the  diam- 
eters of  the  gear  and  pinion.  Another  rule  that  should  be  ob- 
served is  that  the  ratio  of /to  Zi  should  not  exceed  one-third.  For 
values  of  the  permissible  stress  S  for  the  various  grades  of  cast 
materials,  equation  (330)  and  Table  72  should  be  used.  >To 
facilitate  the  use  of  (364),  the  values  of  the  coefficient  m  for 
various  ratios  of  /  to  li  are  put  into  the  form  of  a  graph,  shown  in 
Fig.  161. 

241.  Strength  of  Cut  Teeth. — The  formula  generally  adopted 
by  designers  for  calculating  the  strength  of  cut  bevel  teeth  is  the 
one  proposed  by  Mr.  Wilfred  Lewis.  The  assumptions  regarding 
the  distribution  of  the  tooth  pressure  made  in  the  preceding 
article  will  also  hold  in  the  discussion  of  the  cut  teeth;  hence,  the 
equations  (359)  to  (362)  inclusive  will  hold  in  the  present  case. 
As  in  the  analysis  of  the  cut  spur  gears,  Mr.  Lewis  considered  the 


334 


STRENGTH  OF  CUT  TEETH 


[CHAP.  XIII 


tooth  as  equivalent  to  a  beam  of  uniform  strength,  that  is,  one 
having  a  parabolic  cross-section  as  indicated  in  Fig.  162.  From 
the  geometry  of  the  figure,  it  is  evident  that 

t\  =  4  hlXl 

and  substituting  this  value  in  (362),  and  multiplying  and  dividing 
through  by  p',  we  get 

(365) 


FIG.   162. 


Now  the  ratio  of  2xi  to  3p'  is  simply  the  so-called  Lewis  factor 
discussed  in  detail  in  Art.  223 ;  hence,  replacing  it  by  the  symbol 

T         /        /2  1 
y  and   denoting   the   factor      1  —  ---  H — -^     by  the  symbol   n, 

LI         oli 

equation  (365)  may  be  written 


W,  =  Sp'fyn 


(366) 


As  stated  in  the  discussion  of  cast  teeth,  the  ratio  of  /  to  l\ 
should  not  exceed  one-third  and  the  face  of  the  gear  is  usually 
from  two  to  three  times  the  circular  pitch.  Equation  (330)  and 
the  data  contained  in  Table  72  should  be  used  for  arriving  at 
the  permissible  fiber  stress  for  the  given  material  and  given  con- 
dition of  operation.  It  is  important  to  note  that  the  coefficient 
n  represents  the  ratio  that  the  strength  of  the  bevel  gear  bears  to 
the  strength  of  a  spur  gear  of  the  same  face  and  pitch.  The 


ART.  241] 


STRENGTH  OF  CUT  TEETH 


335 


graph  given  in  Fig.  163  shows  the  relation  existing  between  n  and 
the  ratio  of  /  to  li,  and  will  serve  as  a  time  saver  in  the  solution  of 
bevel  gear  problems. 

The  proportions  of  cut  bevel  teeth  for  the  various  systems  in 
general  use  are  the  same  as  those  given  in  Table  71  and  in  Art. 
230(e). 


n  Q 

N 

^ 

,. 

•v 

s, 

s, 

V 

E 

c 

^ 

(J-O 

^ 

s 

+• 

\^ 

c 

V 

O 

1 

< 

\ 

.' 

V 

^~ 

s. 

_ 

^^ 

V 

X. 

^^ 

iv 

_                    s 

J 

J 

-,       A 

0  6 

:  j 

( 

3 

0 

1 

R 

a 

t 
F 

i 

1C 

0 
}. 

1 

c 

() 

0 
i 

3. 

2 

f 

t 

0 

0 

3                        0,4 

242.  Method  of  Procedure  in  Problems. — In  order  to  save 
time,  the  following  method  may  be  used  in  determining  the 
strength  of  cut  bevel  gear  teeth. 

(a)  Since  y  is  the  form  factor,  its  magnitude  cannot  be  based 
upon  the  actual  number  of  teeth  in  the  gear,  but  must  be  based 
upon  the  so-called  formative  number  of  teeth  as  explained  in 
Art.  235.  To  obtain  the  formative  number  of  teeth  in  the  pinion 
shown  in  Fig.  164,  multiply  the  actual  number  of  teeth  by  the 

2  / 
ratio  -g-.     For  the  gear,  the  formative  number  is  equal  to  the 

2  / 
ratio  —^  multiplied  by  the  actual  number  of  teeth  in  the  gear. 

From  Tables  69  and  70,  determine  the  factor  y  corresponding  to 
the  formative  number. 

(6)  From  Fig.  163  determine  the  magnitude  of  the  coefficient 
n  for  the  assumed  ratio  of  /  to  lit 


336 


RESULTANT  TOOTH  PRESSURE          [CHAP.  XIII 


(c)  For  the  given  material  and  the  speed  of  the  gears  determine 
the  magnitude  of  the  stress  S. 

(d)  Knowing  p'  and  /  and  having  established  values  for  y,  n 
and  Sj  the  load  transmitted  by  either  the  gear  or  pinion  may  be 
calculated. 

243.  Resultant  Tooth  Pressure. — The  formulas  derived  in 
Arts.  240  and  241,  instead  of  giving  the  resultant  load  on  the  gear 
tooth,  merely  give  an  equivalent  load  at  the  large  end  of  the  tooth. 
The  resultant  normal  tooth  pressure,  as  well  as  its  point  of  appli- 
cation, must  be  determined  before  it  is  possible  to  analyze  the 
bearing  loads  and  thrusts  due  to  the  action  of  bevel  gears.  Re- 


FIG.  164. 

f erring  to  the  bevel-gear  tooth  shown  in  Fig.  162,  the  normal  tooth 
pressure  is  considered  as  acting  along  the  outer  edge  of  the  tooth 
as  shown  in  the  end  view  of  the  tooth.  The  line  of  action  of  the 
normal  pressure  intersects  the  center  line  of  the  tooth  at  the  point 
A,  at  a  distance  hi  above  the' weakest  section  of  the  tooth. 

To  determine  the  magnitude  W  of  the  resultant  of  all  the  ele- 
mentary tooth  pressures  dW,  as  well  as  the  point  of  application 
of  this  resultant,  proceed  as  follows: 

From  (357)  it  is  evident  that 

W  =  CS   fllldl  -  —  (g  -  g)  (367) 

Jh  2 

Taking  moments  of  dW  about  the  apex  0,  we  get 
dM  =  IdW  =  CSl2dl 


whence 


(368) 


ART.  243] 


RESULTANT  TOOTH  PRESSURE 


337 


The  distance  that  the  point  of  application  of  W  is  from  the 
apex  0  is  found  by  dividing  M  by  W;  hence 


M      2  r  ll  +  iA  + 

'TF    3 


To  simplify  the  determination  of  Z0,  it  is  best  to  put  (369)  in 

terms  of  the  radius  RQ  shown  in  Fig.  165.  From  the  geometry 
of  the  figure,  it  follows  that 

RQ  =  IQ  sin  a  (370) 


FIG.  165. 


Substituting  in  (369)  the  value  of  I*  =  h  —  f,  and  reducing,  we 
get 


3  sin 


Combining  (370)  and  (371),  we  have 


(371) 


2 


D! 


in  which  Z  denotes  the  factor  » 

O 


(372) 


338 


BEARING  PRESSURES  AND    THRUSTS   [CHAP.  XIII 


To  facilitate  the  use  of  the  formula  for  R0,  the  coefficient  Z  was 
determined  for  various  values  of  the  ratio  /  to  li.  These  values 
were  then  plotted  in  the  form  of  a  graph,  as  shown  in  Fig.  166. 
By  means  of  the  graph  and  (372),  the  value  of  RQ  may  easily  be 
calculated,  since  the  angle  a  is  known  for  any  particular  gear. 

Now  the  magnitude  of  the  resultant  tooth  pressure  W  can  be 
calculated  by  means  of  (367) ,  but  since  the  latter  is  more  or  less 
involved  a  more  direct  method  for  finding  W  is  desirable.  This  is 


I —   0.48 


0.47 


0.46 


0.4S 


£0.44 


0.41 


0.40 


0,1 


02 

Ratio 
FIG.   166. 


0.3 
of    f  to  I 


OA 


obtained  by  dividing  the  torsional  moment  T  on  the  gear  by  the 
radius  RQ',  whence 

W  =  f-  (373) 

It/0 

244.  Bearing  Pressures  and  Thrusts. — Having  determined  the 
resultant  tooth  pressure  W  as  well  as  its  point  of  application,  we 
are  now  prepared  to  discuss  the  pressures  and  thrusts  coming 
upon  the  bearings  of  the  supporting  shaft.  Letting  Wn  in  Fig. 
165  represent  the  resultant  normal  tooth  pressure;  then  resolving 
Wn  along  the  tangent  to  the  pitch  circle,  we  get  the  resultant 
tangential  tooth  pressure 

W  =  FnCOS/3  (374) 


ART.  245]  GEAR-WHEEL  PROPORTIONS  339 

The  component  of  Wn  at  right  angles  to  the  element  of  the  pitch 
cone,  namely,  that  along  the  line  AB  in  Fig.  165,  is 

Wr  =  W  nsin£  =  W  tan/3  (375) 

The  component  W  produces  a  lateral  pressure  upon  the  sup- 
porting bearings  but  no  thrust  along  the  shaft  of  the  gear.  The 
component  Wr  produces  both  lateral  pressure  and  end  thrust, 
the  magnitudes  of  which  are  given  by  the  following  expressions: 

Lateral  pressure  due  to  Wr  =  Wr  cos  a  =  W  tan  (3  cos  a 
Thrust  due  to  Wr  =  Wr  sin  a  =  W  tan  0  sin  a 

To  obtain  the  resultant  lateral  pressure  upon  the  bearings,  the 
two  separate  components  must  be  combined,  either  algebraically 
or  graphically,  and  in  order  to  arrive  at  the  exact  distribution  of 
the  resultant  pressure,  the  location  of  the  bearings  relative  to  the 
gear  must  be  established. 

Graphical  methods  may  also  be  employed  to  determine  W, 
Wrj  and  their  various  components,  as  shown  in  Fig.  165. 


BEVEL-GEAR  CONSTRUCTION 

In  general,  the  constructive  features  of  bevel  gears  are  similar 
to  those  used  for  spur  gears.  Small  pinions  are  made  solid  as 
shown  in  Fig.  158,  and  for  economy  of  material  larger  pinions 
are  made  with  a  web.  Examples  of  the  latter  construction  are 
shown  in  Figs.  158  and  159.  Not  infrequently  the  webs  are  pro- 
vided with  holes  in  order  to  decrease  the  weight  of  such  gears. 
Large  bevel  gears  are  made  with  arms,  the  design  of  which  will  be 
discussed  in  the  following  article.  Bevel  gears  are  seldom  made 
in  extremely  large  sizes,  and  for  that  reason  split  or  built-up 
gears  are  used  but  little. 

245.  Gear-wheel  Proportions.— (a)  Arms. — In  bevel  gears  the 
T-arm  is  remarkably  well  adapted  for  resisting  the  stresses  that 
come  upon  it,  and  for  that  reason  is  used  rather  extensively  in 
gears  of  large  size.  In  small  gears,  however,  the  greater  cost  of 
the  arm  construction  more  than  offsets  the  saving  of  material; 
therefore  for  such  gears  the  web  and  solid  centers  are  in  common 
use.  Fig.  167  shows  a  bevel  gear  with  a  T-arm. 

The  rib  at  the  back  of  the  arm  is  added  to  give  lateral  stiffness, 
that  is,  to  take  care  of  the  load  component  Wr  discussed  in  Art. 
244.  This  rib  adds  practically  nothing  to  the  resistance  of  the 


340 


GEAR-WHEEL  PROPORTIONS 


[CHAP.  XIII 


arm  to  bending  in  the  plane  of  the  wheel,  and  for  that  reason,  in 
deriving  the  formula  for  the  strength  of  the  arm,  the  effect  of  the 
rib  is  not  considered.  As  in  the  case  of  spur  gears,  the  arm  is 
treated  as  a  cantilever  beam  under  flexure,  and  it  is  assumed  that 
each  arm  will  carry  its  proportionate  share  of  the  load  trans- 
mitted by  the  gear. 

Denoting  the  thickness  and  width  of  the  arm  by  b  and  h, 
respectively,  and  equating  the  external  moment  to  the  resisting 
moment,  we  get 

TfiDi       Sbh2 


2n 


6 


in  which  n  denotes  the  number  of  arms,  and  W\  and  Di  are  the 


equivalent  load  and  pitch  diameter,  respectively,  at  the  large  end 
of  the  tooth. 

Solving  for  h,  we  have 

fifrr^  (377) 


h  = 

The  dimension  b  is  generally  made  equal  to  about  one-half  of 
the  circular  pitch,  as  shown  in  Fig.  167.  The  permissible  stress 
for  cast  iron  varies  from  1,500  to  3,000  depending  upon  the  size 
of  the  gear.  The  thickness  of  the  rib  on  the  back  of  the  arm 
proper  is  made  as  shown  in  the  figure. 


ART.  247]  MOUNTING  BEVEL  GEARS  341 

(6)  Rim  and  hub. — For  the  proportions  of  the  rim  and  the  rein- 
forcing bead  on  the  inside  of  the  rim,  consult  Fig.  167.  The  hub 
is  made  similar  to  those  used  for  spur  gears,  proportions  of  which 
are  given  in  Table  74. 

246.  Non-metallic  Bevel  Gears. — Frequently  where  noiseless 
operation  is  desirable  bevel  gears  made  of  rawhide  and  Fabroil 
are  used.     In  Fig.  141(6)  is  shown  the  design  of  a  rawhide  gear 
that  has  given  excellent  service.     The  same  general  constructive 
feature  would  be  used  when  a  Fabroil  filler  is  employed ;  but  in 
place  of  the  plain  rivets,  the  threaded  type  should  be  used,  as 
recommended  by  the  manufacturer  of  such  gears.     In  general, 
the  discussion  of  non-metallic  gears  given  in  the  preceding  chapter 
applies  also  to  bevel  gears. 

247.  Mounting  Bevel  Gears. — To  obtain  good  service  from  an 
installation  of  bevel  gears,  it  is  important  that  the  material  used 
for  the  pinion  and  gear  be  chosen  with  some  care  and  that  the 
teeth  be  formed  and  cut  accurately.     These  two  factors  alone, 
however,  do  not  necessarily  make  a  successful  drive,  as  poorly 
designed  mountings  are  frequently  the  source  of  many  bevel- 
gear  failures.     The  following  important  points  should  be  observed 
in  designing  the  mountings  of  a  bevel-gear  drive: 

1.  Make  the  bearings  and  their  .supports  rigid,  and  so  that  all 
parts  may  be  easily  assembled. 

2.  Make  provisions  for  taking  care  of  the  end  thrust  caused  by 
the  component  Wr  discussed  in  Art.  244. 

3.  Make  provisions  for  lubricating  the  bearings  and  if  neces- 
sary the  gears  themselves. 

4.  Provide  the  gears  with  a  dustproof  guard,  thus  protecting 
the  gears  and  at  the  same  time  protecting  the  operator  of  the 
machine. 

5.  The  shafts  supporting  the  gears  should  be  made  large,  so  as 
to  provide  the  necessary  rigidity.     Slight  deflections  of  bevel-gear 
shafts  produce  noisy  gears  and  cause  the  teeth  to  wear  rapidly. 

(a)  Solid  bearing.- — A  rigid  construction  used  to  a  considerable 
extent  on  machine  tools  is  the  solid  bearing  construction',  two 
designs  of  which  are  shown  in  Figs.  168  and  169.  In  both  of 
these  designs  the  end  thrusts  are  taken  care  of  by  the  use  of 
bronze  washers,  as  shown.  The  bearings  throughout  are  bronze 
bushed.  The  type  of  bevel-gear  drive  illustrated  in  Fig.  169  is 
used  when  the  pinion  is  splined  to  its  shaft.  In  such  cases  the 


342 


MOUNTING  BEVEL  GEARS 


[CHAP.  XIII 


hub  of  the  pinion  is  made  long,  so  that  it  may  serve  as  a  bearing. 
The  heavy  thrust  is  taken  care  of  by  the  self-aligning  steel 
washers,  between  which  is  located  one  made  of  bronze.  In  place 


FIG.   168. 


of  the  bronze  thrust  washers  shown  in  Figs.  168  and  169,  ball 
thrust  bearings  may  be  used.  The  latter  type  of  bearings  are 
more  expensive  than  the  bronze  washers,  and  unless  they  are 


FIG.   169. 


designed  correctly  they  are  liable  to  be  troublesome.  Of  late,  the 
type  of  radial  ball  bearing  that  is  capable  of  taking  a  certain 
amount  of  thrust,  in  addition  to  the  transverse  load,  is  being 


ART.  247] 


MOUNTING  BEVEL  GEARS 


343 


used  in  connection  with  bevel-gear  drives.     The  conical  roller 
bearing  is  also  adapted  for  use  with  bevel-gear  transmissions. 

(6)  Ball  bearing. — In  Fig.  170  is  shown  a  design  of  a  bevel-gear 
drive  in  which  ball  bearings  are  used  throughout.  This  form  of 
drive  is  used  on  a  drill  press  and  the  details  were  worked  out  by 
The  New  Departure  Mfg.  Co.,  makers  of  ball  bearings.  The 
double-row  ball  bearings  take  both  radial  loads  and  thrusts, 
while  the  single-row  ball  bearing  having  a  floating  outer  race 
takes  only  a  transverse  load.  The  double-row  ball  bearing  on 
the  horizontal  driving  shaft  is  mounted  in  a  shell  or  housing 
which  is  adjustable,  thus  providing  means  for  getting  the  proper 
tooth  engagement  between  the  pinion  and  the  gear.  Necessarily, 


FIG.  170. 

this  form  of  construction  will  call  for  a  bearing  having  a  floating 
outer  race  at  the  farther  end  of  the  drive  shaft. 

When  it  is  desired  to  support  the  bevel  pinion  between  two 
bearings,  the  design  shown  in  Fig.  171  will  give  good  results.  The 
drive  illustrated  in  this  figure  is  one  that  is  used  on  the  rear  axle 
of  an  automobile.  The  arrangement  and  selection  of  the  various 
bearings  were  worked  out  by  the  Gurney  Ball  Bearing  Co.  The 
duplex  bearing  back  of  the  pinion,  having  a  thrust  capacity  of 
one  and  one-half  times  the  radial  load,  is  mounted  rigidly  in  an 
adjustable  cage.  The  bearing  at  the  other  end  of  the  pinion  shaft 
is  of  the  radial  type  and,  as  shown,  is  mounted  so  as  to  permit  a 
movement  lengthwise  of  the  shaft.  The  advantage  of  using  the 


344 


MOUNTING  BEVEL  GEARS 


[CHAP.  XIII 


cage  construction  just  mentioned  is  that  the  pinion  with  its  shaft 
and  bearings  may  be  assembled  on  the  bench  as  a  unit.  The 
bearing  to  the  left  of  the  bevel  gear  is  of  a  type  capable  of  taking 
a  thrust  equal  to  the  transverse  load.  The  bearing  supporting 
the  other  end  of  the  differential  housing  to  which  the  bevel  gear 
is  fastened,  is  also  of  the  combined  radial  thrust  type;  but  in  this 
case  the  thrust  capacity  is  equivalent  to  one-half  of  the  radial 
load.  In  Fig.  171,  the  differential  bevels  and  the  two  axles  are 
not  shown,  in  order  to  bring  out  more  clearly  the  other  important 


FIG.  171. 

details.  The  type  of  bevel  gearing  used  in  the  design  just  dis- 
cussed is  the  so-called  " spiral  bevel"  which  will  be  discussed  in 
the  following  article. 


SPECIAL  TYPES  OF  BEVEL  GEARS 

248.  Spiral  Bevel  Gears. — A  special  type  of  bevel  gears  called 
"spiral  bevels"  is  now  used  extensively  for  driving  the  rear  axles 
of  automobiles.  No  doubt  within  a  short  time  manufacturers  of 
machine  tools  and  other  classes  of  machinery  will  begin  to  use 


ART.  2491 


SPIRAL  BEVEL  GEARS 


345 


spiral  bevels,  since  they  possess  certain  advantages  over  the 
straight-tooth  gears.  The  teeth  of  these  gears  are  curved  on  the 
arc  of  a  circle  if  produced  by  the  well-known  Gleason  spiral 
bevel-gear  generator,  or  they  are  helical  if  produced  on  a  generat- 
ing-gear  planer.  An  illustration  of  the  former  type  is  shown  in 
Fig.  172. 

In  discussing  spiral  bevel  gears,  one  should  be  familiar  with 
certain  terms  or  expressions  that  are  now  in  common  use.  These 
are  as  follows: 

(a)  Angle  of  spiral. — By  the  angle  of  spiral  is  meant  the  angle 
that  the  tangent  AB  to  the  tooth  at  the  center  of  the  gear  face 


FIG.  172. 

makes  with  the  element  OA  of  the  pitch  cone.  In  Fig.  172  this 
angle  is  designated  by  the  symbol  a. 

(6)  Direction  of  spiral. — The  direction  of  the  spiral  is  desig- 
nated as  right  or  left  hand,  based  upon  the  direction  of  the  spiral 
on  the  pinion ;  thus,  by  left-hand  spiral  is  meant  left  hand  on  the 
pinion  and  right  hand  on  the  gear. 

(c)  Lead. — By  the  term  lead  is  meant  the  distance  that  the 
spiral  advances  within  the  face  of  the  gear,  as  shown  in  Fig.  172. 

249.  Advantages  and  Disadvantages. — (a)  Advantages. — Among 
the  advantages  claimed  for  spiral  and  helical  bevel  gears  are  the 
following : 


346  SPIRAL  BEVEL  GEARS  [CHAP.  XIII 

1.  Due  to  the  curvature  of  the  teeth  their  engagement  is  grad- 
ual, thus  tending  to  eliminate  noise.     The  best  results,  accord- 
ing to  the  Gleason  Works,  are  obtained  when  the  lead  of  the  spiral 
is  made  equal  to  one  and  one-quarter  to  one  and  one-half  times 
the  pitch  of  the  teeth. 

2.  The  wear  on  the  teeth  of  spiral  bevel  gears  is  no  more  than 
on  the  teeth  of  the  common  type  of  bevel. 

3.  It  has  been  found  in  practice  that  spiral  bevel  pinions  permit 
of  greater  endwise  adjustment  than  straight-tooth  bevels,  with- 
out producing  excessive  noise  or  causing  bearing  troubles. 

4.  There  is  practically  no  difference  between  the  load-carrying 
capacity  of  spiral  and  helical  bevel  gears,  when  compared  with 
those  having  straight  teeth. 

5.  Spiral  and  helical  bevel  gears  are  better  adapted  to  high- 
gear  ratios,  5  and  6  to  1  giving  satisfactory  service,  while  with 
straight  teeth  4J-^  to  1  seems  to  be  about  the  dividing  line  between 
quiet  and  noisy  gears  when  run  at  high  speeds  such  as  are  com- 
mon in  automobile  transmissions. 

(b)  Disadvantages. — The  chief  disadvantages  resulting  from 
the  use  of  spiral  or  helical  bevel  gearing  is  the  provision  that  must 
be  made  to  take  care  of  the  additional  thrust  coming  upon  the 
bearings.  In  installations  where  the  direction  of  rotation  is 
reversed,  the  end  thrust  on  the  bearing  must  be  taken  care  of  in 
both  directions,  as  the  analysis  given  in  the  following  article  will 
show.  Due  to  the  additional  end  thrust  on  the  bearing,  it  is 
probable  that  the  efficiency  of  a  spiral  bevel-gear  drive  is  slightly 
less  than  that  obtained  from  a  common  bevel-gear  drive. 

250.  Bearing  Loads  and  Thrusts. — The  following  analysis, 
applied  to  the  spiral  bevel  gear,  is  based  on  the  assumption  that 
this  form  of  tooth  may  be  treated  in  a  manner  similar  to  a  straight 
tooth  having  a  spiral  angle  equal  to  the  spiral  angle  measured  at 
the  center  of  the  face,  as  defined  in  Art.  248.  Furthermore,  the 
friction  of  tooth  contact  will  not  be  considered.  To  arrive  at 
expressions  for  the  bearing  loads  and  thrusts,  proceed  as  follows : 

(a)  Direct  rotation. — The  spiral  bevel  gear,  shown  in  Fig.  173, 
has  an  angle  of  spiral  designated  by  a,  and  an  angle  of  obliquity 
of  tooth  pressure  equal  to  /?.  Resolving  the  resultant  normal 
tooth  pressure,  acting  at  G  and  represented  by  the  vector  AB, 
into  three  components,  we  have: 

1.  The  component  DF  perpendicular  to  the  plane  of  the  paper 


ART.  250]  SPIRAL  BEVEL  GEARS  347 

and  also  equal  to  W,  the  tangential  force  acting  on  the  gear  at  Gt 
is  given  by  the  following  expression : 

W  =  ABcosa  cos/3  (378) 

2.  The  component  acting  along  the  element  of  the  pitch  cone  is 
represented  by  EF  and  its  magnitude  is 

EF  =  HG  =  TFtana  (379) 

3.  The  component  at  right  angles  to  the  element  of  the  pitch 


FIG.  173. 

cone  is  represented  by  the  vector  BC  or  GI,  the  magnitude  of 
which  is 

BC  =  GI  =  ABsmp 


=  w^P 
cos  a 


(380) 


Resolving  the  three  forces  DF,  HG,  and  GI  into  components 
whose  lines  of  action  are  along  the  center  line  of  the  shaft  and  at 
right  angles  thereto  we  obtain  for  the  thrust  along  the  shaft  of 
the  gear 

Fy  =  HGcos  B  +  GIsm  6 

W 

=  -^—  (sin  «  cos  9  +  tan  0  sin  0)  (381) 


348  SPIRAL  BEVEL  GEARS  [CHAP.  XIII 

and  for  the  thrust  along  a  line  at  right  angles  to  the  shaft  of  the 
gear,  or  in  other  words  along  the  shaft  of  the  pinion 

Fx  =  HG  sin  0  -  GIcos  d 

W 

=  -     -  (sin  a  sin  6  -  tan  0  cos  6)  (382) 

cos  a 

It  follows  that  the  thrust  exerted  by  the  pinion  upon  its  shaft  has 
a  magnitude  given  by  (382),  but  its  direction  is  opposite  to  that 
of  Fx. 

(b)  Reversed  rotation. — Supposing  now  that  the  direction  of 
rotation  of  the  gear  is  reversed,  the  component  along  the  element 
of  the  cone,  given  by  (379),  reverses  in  direction,  or  in  other 
words,  it  acts  toward  the  point  0  in  Fig.  173;  thus 

EF  =  GH  =      -  W  tana  (383) 

Furthermore,  the  component  BC  or  GI  at  right  angles  to  the  cone 
element  remains  as  in  the  preceding  case. 

Resolving  DF,  GH,  and  GI,  as  in  the  preceding  case,  we  get  for 
the  thrust  along  the  shaft  of  the  gear 

Fy  =  GI  sin  B  +  GH  cos  B 

W 

-  (tan  |3  sin  6  —  sin  a  cos  6)  (384) 

In  a  similar  manner,  the  magnitude  of  the  thrust  along  the 
pinion  shaft  is  found  to  be 

Fx  =  HG  sin  0  -  GI  cos  0 
W 


cos  a 


(sin  6  sin  a  +  tan  0  cos  B)  (385) 


If  the  spiral  of  the  teeth  is  reversed  for  the  case  just  discussed, 
the  equations  deduced  for  the  preceding  case  will  hold. 

251.  Experimental  Results. — In  order  to  determine  the  actual 
thrusts  upon  the  bevel  pinion  of  automobile  drives,  the  Gleason 
Works  made  an  extensive  series  of  tests  upon  various  types  of 
bevel  gears.  The  results  were  published  in  Machinery,  vol.  20, 
p.  690.  Table  78  gives  the  various  dimensions  and  angles  of  the 
gears  and  pinions,  and  the  average  pinion  thrusts  per  100  pounds 
of  load  on  the  tooth.  The  pinion  thrusts  have  been  calculated  by 
substituting  in  (382)  and  (385)  the  value  of  W  and  the  values 
of  the  functions  of  the  various  angles.  Comparison  of  these 


ART.  252] 


SKEW  BEVEL  GEARS 


349 


calculated  values  with  the  actual  pinion  thrusts  observed  in  the 
tests  show  good  agreement. 

TABLE   78. — EXPERIMENTAL   DATA   PERTAINING  TO   BEVEL  GEARING 


Type  of  bevel  gearing 

Common 

Spiral  tooth 

1 

2 

j  Number  of  teeth 

f  Pinion 
{  Gear 

15 
53 

14 
53 

15 
53 

3 

Pressure  angle  —  j(3 

14^  degrees 

4 
5 

j  Pitch  cone  angle  —  d 

f  Pinion 
I  Gear 

15°-48' 
74°-12' 

14°-48' 
75°-12' 

15°-48' 
75°-12' 

6 

Spiral  angle  —  a 

0 

19°-45' 

31°-21' 

7 

Pinion     thrust  f  Direct 

f  Actual 

7.34 

-28.70 

-49.50 

8 

in  pounds   per  {    drive 

\  Calculated 

7.06 

-27.6 

-50.3 

9 

100  pounds  of  f  Reverse 

f   Actual 

7.62 

45.00 

73.82 

10 

tooth  load           {  drive 

{  Calculated 

7.06 

41.6 

66.9 

252.  Skew  Bevel  Gears. — Another  form  of  special  bevel  gear, 
known  as  a  skew  bevel  gear,  has  no  common  axes  plane,  and  hence 
the  face  and  cutting  angles  of  the  pinion  and  gear  do  not  converge 
to  a  common  apex.  This  fact  introduces  more  or  less  involved 
mathematical  calculations  in  arriving  at  the  various  angles  re- 
quired to  lay  out  such  gears.  Because  of  the  more  involved 
calculations  required  and  the  greater  cost  of  manufacture,  skew 
bevel  gears  are  rarely  used  in  machine  construction.  Strictly 
speaking,  there  are  two  distinct  types  of  skew  bevel  gears,  as 
follows:  (1)  Those  in  which  the  oblique  teeth  are  confined  to  the 
gear,  and  the  mating  gear  or  pinion  is  really  a  straight  tooth 
bevel;  (2)  those  in  which  the  teeth  of  both  gear  and  pinion  are 
oblique. 

References 

American  Machinist  Gear  Book,  by  C.  H.  LOGUE. 
A  Treatise  on  Gear  Wheels,  by  G.  B.  GRANT. 
Spur  and  Bevel  Gearing,  by  Machinery. 
Elements  of  Machine  Design,  by  J.  F.  KLEIN. 
Constructeur,  by  REULEAUX. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 
Bearing  Pressures  Due  to  the  Action  of  Bevel  Gears  under  Load,  Mchy., 
vol.  20,  p.  639. 

Gleason  Spiral  Type  Bevel  Gear  Generator,  Mchy.,  vol.  20,  p.  690. 
Spiral  Type  Bevel  Gears,  Mchy.,  vol.  23,  p.  199. 
Laying  out  Skew  Bevel  Gears,  Mchy.,  vol.  23,  p.  32. 


CHAPTER  XIV 
SCREW  GEARING 

The  term  screw  gearing  is  applied  to  all  classes  of  gears  in 
which  the  teeth  are  of  screw  form.  Screw  gearing  is  used  for 
transmitting  power  to  parallel  shafts  as  well  as  to  non-parallel  and 
non-intersecting  shafts.  The  following  two  classes  of  screw 
gearing  are  used  considerably  in  machine  construction:  (a) 
helical  gearing;  (6)  worm  gearing. 


HELICAL  GEARING 


253.  Types  of  Helical  Gears. — Helical  gearing  may  be  used  for 
the  transmission  of  power  to  shafts  that  are  parallel,  or  to  shafts 


FIG.  174. 

that  are  at  right  angles  to  each  other  and  do  not  intersect,  or  to 
shafts  that  are  inclined  to  each  other  and  do  not  intersect.  The 
teeth  of  helical  gears  used  for  connecting  shafts  that  are  parallel 
have  line  contact,  while  those  used  for  connecting  non-parallel, 
non-intersecting  shafts  have  merely  point  contact  and  for  that 
reason  are  not  used  much  for  the  transmission  of  heavy  loads. 
From  Fig.  174,  it  is  evident  that  the  normal  component  of  the 
tangential  load  W  on  the  teeth  of  a  pair  of  helical  gears  connect- 
ing two  parallel  shafts  produces  an  end  thrust  on  each  shaft.  To 

350 


ART.  254] 


HELICAL  GEARING 


351 


overcome  this  objectionable  end  thrust,  two  single  helical  gears 
having  teeth  of  opposite  hand  are  sometimes  bolted  or  riveted 
together,  forming  what  is  called  the  double-helical  or  herringbone 
gear.  Due  to  improved  methods  of  cutting  helical  teeth,  herring- 
bone gears  are  not  now  constructed  to  any  great  extent  from  two 
single-helical  gears,  but  are  cut  directly  from  the  solid  blank. 
Herringbone  gears  are  also  produced  by  casting  them  in  a  prop- 
erly constructed  mould. 

There  are  two  general  types  of  double-helical  gears,  as  follows : 

(a)  The  ordinary  herringbone  gear  in  which  the  two  teeth  meet 

at  a  common  apex  at  the  center  of  the  face,  as  shown  in  Fig. 

175 (a).     A  modification  of  this  type,  in  which  the  central  part 

has  been  removed,  is  shown  in  Fig.  175(6). 


(a) 


(b) 

FIG.  175. 


(b)  The  type  known  as  the  Wuest  gear  in  which  the  teeth 
instead  of  coming  together  at  a  common  apex  at  the  center  of  the 
face  do  not  meet  at  all,  but  are  staggered  as  shown  in  Fig.  175(c). 

In  the  types  illustrated  by  Fig.  175(6)  and  (c),  a  groove  is 
turned  into  the  face  as  shown,  so  as  to  provide  clearance  for  the 
cutters  used  in  cutting  the  teeth.  In  gears  having  teeth  cast 
approximately  to  shape,  the  center  part  where  the  two  teeth  come 
together  is  cast  somewhat  undersize  on  both  sides  of  the  teeth, 
also  at  the  bottom  of  the  space  between  the  teeth  as  shown  in 
Fig.  175(a). 

254.  Advantages  of  Double -helical  Gears. — When  com- 
pared with  a  spur  gear,  a  double-helical  gear  has  the  following 
advantages : 

(a)  The  face  of  the  gear  is  always  made  long  so  that  more  than 
one  tooth  is  in  action;  in  other  words,  the  continuity  of  tooth 


352  APPLICATIONS  OF  HELICAL  GEARING    [CHAP.  XIV 

action  depends  upon  the  face  of  the  gear  and  not  upon  the  num- 
ber of  teeth  in  the  pinion  as  with  spur  gearing. 

(6)  Due  to  the  continuity  of  action,  the  load  is  transferred 
from  one  tooth  to  another  gradually  and  without  shock,  thus 
eliminating  to  a  great  extent  noise  and  vibration. 

(c)  In  helical  gearing,  the  load  is  distributed  across  the  face  of 
the  gear  along  a  diagonal  line,  thus  decreasing  the  bending  stress 
in  the  teeth. 

(d)  In  well-designed  helical  gearing  all  phases  of  engagement 
occur  simultaneously,   hence   the  load  is  transmitted  by  sur- 
faces that  are  partly  in  sliding  contact  and  partly  in  rolling  con- 
tact.    Such  action  has  a  tendency  to  equalize  the  wear  all  over 
the  teeth,  consequently  the  tooth  profile  is  not  altered. 

(e)  Actual  tests  on  double-helical  gears  show  that  they  have 
much  higher  efficiencies  than  those  obtained  from  spur  gears. 
Efficiencies  of  98  to  99  per  cent,  are  not  unusual  with  properly 
designed  transmissions. 

(/)  Gear  ratios  much  higher  than  those  used  with  spur  gearing 
may  be  employed. 

(g)  Due  to  the  absence  of  noise  and  vibration,  double-helical 
gears  may  be  run  at  much  higher  pitch  line  speeds  than  is  pos- 
sible with  spur  gearing. 

255.  Applications  of  Double -helical  Gears. — Cut  double-heli- 
ical  gears  have  been  applied  successfully  to  many  different  classes 
of  service.  The  following  examples  of  applications  give  some 
idea  of  the  extent  of  the  field  in  which  such  gears  may  be  used. 

(a)  Drives  for  rolling  mills. — Gears  used  for  driving  rolling  mills 
operate  under  very  unfavorable  conditions,  such  as  heavy  over- 
loads, the  magnitudes  of  which  are  difficult  to  determine;  further- 
more, these  overloads  are  applied  suddenly  and  are  constantly 
repeated.  The  gears  are  also  subject  to  excessive  wear  due  to  the 
dirty  surroundings.  Double-helical  gears  are  now  installed  for 
rolling-mill  drives,  and,  due  to  the  continuous  tooth  engagement, 
such  gears  readily  withstand  the  suddenly  applied  loads.  When- 
ever it  is  possible,  the  gears  should  be  enclosed  by  a  casing  and 
run  in  oil,  thereby  eliminating  all  noise. 

(6)  Drives  for  reciprocating  machinery. — Gears  for  motor-driven 
reciprocating  pumps  and  air  compressors  are  required  to  transmit 
a  torsional  moment  which  varies  between  rather  wide  limits,  sev- 
eral times  per  revolution.  Due  to  the  load  fluctuation,  an  or- 
dinary spur-gear  drive  is  noisy  and  is  subject  to  considerable 


ART.  2£6]  HELICAL  GE'AR  TOOTH  SYSTEMS  353 

vibration,  while  a  double-helical  gear  drive  runs  quietly,  without 
vibration,  and  at  the  same  time  is  more  efficient. 

(c)  Drives  for  hoisting  machinery. — In  connection  with  motor- 
driven  hoists  such  as  are  used  in  mines,  double-helical  gears  are 
especially  well  adapted,  since  the  high  gear  ratios  possible  sim- 
plify   the    drives.     High-ratio    double-helical    gears    are    more 
efficient  and  run  more  quietly  than  spur  gears  having  the  same 
ratio.     Such  high-ratio  helical  gears  are  also  being  introduced  on 
modern  high-speed  traction  elevators,  with  excellent  results. 

(d)  Drives  for  machine  tools. — Double-helical  gears  used  on 
motor-driven  machine  tools  produce  a  noiseless  drive  free  from 
vibration,  and  are  better  adapted  to  the  high  speeds  that  are  now 
common  in  machine-tool  drives. 

(e)  Drives  for  steam  turbines. — Gears  used  for  reducing  the  speed 
of  a  steam  turbine  to  that  required  by  a  centrifugal  pump,  fan,  or 
generator  must  be  made  accurately,  as  the  pitch  line  velocity  is 
likely  to  be  from  3,000  to  5,000  feet  per  minute.     Due  to  the  high 
efficiency  and  quiet  running  obtainable  by  the  use  of  double-heli- 
cal gears,  the  latter  are  used  extensively  in  steam-turbine  drives. 
In  such  installations  the  pinions  are  always  made  from  an  alloy- 
steel  forging,  and  after  being  machined  they  are  heat  treated. 

256.  Tooth  Systems. — Several  of  the  more  prominent  manu- 
facturers of  double-helical  gears  agree  fairly  well  on  the  following 
points  relating  to  the  proportions  of  the  teeth  : 

1.  The  tooth  profile  should  be  formed  by  a  20-degree  involute 
curve,  thus  making  the  tooth-pressure  angle  20  degrees. 

2.  The  tooth  should  be  made  shorter  than  the  old  standard 
used  with  spur  gears. 

3.  The  angle  of  the  helix,  more  commonly  called  the  angle  of 
inclination  of  the  tooth,  should  be  23  degrees. 

4.  The  diametral  pitch  standard  should  prevail  for  all  cut  teeth. 

5.  The  unequal  addendum  system  should  be  used  on  all  pinions 
having  few  teeth. 

(a)  Tooth  proportions. — The  proportions  for  the  teeth  and  gear 
blank  given  in  Table  79  are  those  proposed  and  recommended  by 
Mr.  P.  C.  Day  of  The  Falk  Co.  of  Milwaukee,  Wis.  It  should  be 
noted  that  according  to  these  formulas  the  pitch  and  outside 
diameters  of  gears  having  less  than  20  teeth  are  made  slightly 
larger  than  those  of  a  standard  gear.  This  is  done  to  avoid  under- 
cutting of  the  teeth.  If  a  pinion  proportioned  in  this  way 
meshes  with  a  gear  having  less  than  40  teeth,  then  the  distance 


354 


STRENGTH  OF  HELICAL  TEETH  [CHAP.  XIV 


between  the  shafts  must  be  increased  by  an  amount  equal  to  one- 
half  of  the  increase  in  the  pinion  diameter.  If  the  gear,  meshing 
with  a  small  pinion  has  more  than  40  teeth  the  normal  center 

TABLE  79 

1.  Tooth  profile Involute. 

2.  Pressure  angle 20  degrees. 

3.  Angle  of  helix 23  degrees. 

0  8 

4.  Length  of  addendum  =  —  =  0.2546p'. 

5.  Length  of  dedendum  =  --  =  0.3183  p'. 

6.  Full  height  of  tooth     =  —  =  0.5729  p'. 

0  Qpj  T  4-  1 

7.  Pitch  diameter,  when     T  <  20  =  -  — 

T      P 

8.  Pitch  diameter,  when       T  >  20  =  — 

P 

9.  Outside  diameter,  when  T  <  20  = 


10.  Outside  diameter,  when  T  >  20 


T  +  1.6 


distance  may  be  used  by  decreasing  the  pitch  diameter  of  this 
gear  by  the  same  amount  that  the  pinion  diameter  was  increased. 

Gears    made    according    to 
TABLE  80. — PROPORTIONS  OF  TEETH  FOR     . 

CUT  DOUBLE-HELICAL  TEETH,  the  above  suggestions  have 

FAWCUS  MACHINE  Co.  teeth  of  standard  depth  but 

unequal  addendums. 

In  Table  80  are  given  the 
commercial  pitches,  tooth 
proportions,  and  minimum 
lengths  of  face  recommended 
by  the  Fawcus  Machine  Co. 
for  double-helical  gears  hav- 
ing a  pressure  angle  of  20 
degrees  and  a  helix  angle  of 
23  degrees. 

257.  Strength  of  Double- 
helical  Teeth. — Various 
formulas  have  been  pro- 
posed for  determining  the 


Pitch 

Adden- 
dum 

Deden- 
dum 

Minimum 
face 

Dia. 

Cir. 

8.00 

0.393 

0.100 

0.125 

2.5 

6.00 

0.524 

0.133 

0.167 

3.5 

5.00 

0.628 

0.160 

0.200 

4.0 

4.00 

0.785 

0.200 

0.250 

5.0 

3.50 

0.898 

0.229 

0.286 

5.5 

3.00 

1.047 

0.267 

0.333 

6.5 

2.50 

1.257 

0.320 

0.400 

7.5 

2.00 

1.571 

0.400 

0.500 

9.5 

1.75 

1.795 

0.457 

0.572 

11.0 

1.50 

2.094 

0.533 

0.667 

12.5 

1.25 

2.513 

0.640 

0.800 

15.0 

1.00 

3.142 

0.800 

1.000 

19.0 

working  load  that  a  cut 
double-helical  gear  will  transmit;  probably  the  most  reliable  are 
those  given  by  Mr.  W.  C.  Bates  and  Mr.  P.  C.  Day. 


ART.  257]  STRENGTH  OF  HELICAL  TEETH  355 

(a)  Bates'  Formula. — In  an  article  entitled  "The  Design  of 
Cut  Herringbone  Gears,"  published  in  the  American  Machinist, 
Mr.  W.  C.  Bates,  mechanical  engineer  of  the  Fawcus  Machine 
Co.,  proposed  a  formula  for  the  permissible  working  load  for  a 
double-helical  gear,  which  is  really  an  adaptation  of  the  well- 
known  Lewis  spur-gear  formula  given  in  Art.  223.  The  author 
introduces  two  additional  factors,  one  of  which  depends  upon 
the  condition  of  the  load,  whether  it  is  constant  or  variable, 
and  the  second  takes  into  consideration  the  lubrication  neces- 
sary to  prevent  wear.  In  addition  to  these  factors,  higher  fiber 
stresses  than  those  commonly  used  with  the  Lewis  formula 
are  recommended.  The  formula  as  proposed  by  Mr.  Bates  is 
as  follows: 

W  =  X  Sp'fy  CK,  (386) 

in  which  the  factors  p' ',  /,  and  y  have  the  same  meaning  as  assigned 
to  them  in  Art.  223. 

The  factor  C  depends  upon  the  ratio  of  the  maximum  load  to 
the  average  load  during  a  complete  operating  cycle.  If  the 
load  is  fairly  uniform,  that  is,  if  the  ratio  of  maximum  to  average 
load  is  practically  unity,  then  C  is  given  its  maximum  value, 
namely  unity.  If,  however,  the  load  on  the  gear  varies,  say 
from  zero  to  a  maximum  twice  in  a  revolution,  as,  for  example, 
when  the  gear  drives  a  single-cylinder  pump  or  compressor, 
then  C  must  be  given  some  value  less  than  unity.  Experience 
should  dictate  the  magnitude  of  the  factor  C,  and  the  following 
values,  obtained  from  information  furnished  by  Mr.  Bates, 
will  serve  as  a  guide  in  the  selection  of  the  proper  value  for  any 
particular  class  of  service. 

1.  For  reciprocating  pumps  of  the  triplex  type,  C  usually  is 
taken  as  0.7. 

2.  For  mine  hoists  running  unbalanced,  C  is  taken  as  0.57. 

3.  For  rolling  mill  drives  in  which  the  flywheels  are  located  on 
the  pinion  shaft,  the  factor  C  varies  from  0.50  to  0.66,  depending 
upon  the  rapidity  with  which  the  energy  in  the  flywheel  is 
given  up. 

The  factor  K  depends  for  its  value  upon  the  effectiveness  of 
the  lubricating  system  used  with  the  gears;  in  other  words,  the 
wearing  conditions  of  the  gear  depend  upon  K.  When  the  gears 
are  encased  so  that  the  lower  part  of  the  gear  runs  in  oil,  thus 
carrying  a  continuous  supply  of  oil  to  the  mating  pinion,  the  fac- 
tor K  may  be  assumed  as  unity.  It  is  claimed  that  with  such  a 


356 


STRENGTH  OF  HELICAL  TEETH  [CHAP.  XIV 


system  of  lubrication  double-helical  gearing  may  be  operated 
successfully  at  speeds  of  2,000  to  2,500  feet  per  minute.  Experi- 
ence seems  to  indicate  that  with  speeds  exceeding  2,500  feet  per 
minute  considerable  oil  is  thrown  off  the  gears  due  to  centrifugal 
action,  and  in  such  installations  it  is  suggested  that  the  oil,  under 
a  low  pressure,  be  sprayed  against  the  teeth  on  the  entering  side 
near  the  line  of  engagement.  For  other  systems  of  lubrication, 
the  values  of  K  given  in  Table  81  are  recommended. 

TABLE  81. — VALUES  OF  K  AS  RECOMMENDED  BY  W.  C.  BATES 


Bailie  of  K 

Min. 

Max. 

Mean 

Continuous  supply  of  oil  

1 

1 

1 

Thorough  grease  lubrication 

0  83 

0  91 

0  87 

Scanty  lubrication,  but  frequent  inspection  
Indifferent  lubrication.  .  .          

0.80 

0  77 

0.87 
0  83 

0.835 
0  80 

The  permissible  fiber  stress  S  may  be  determined  by  means  of 
the  formula 

1,200 


S  =  So 


1,200  +  V 


(387) 


which  is  similar  in  form  to  the  expression  given  for  the  safe  stress 
in  the  case  of  spur  gearing.  The  values  of  So  given  in  Table  72 
may  also  be  used  for  this  class  of  gearing. 

The  values  of  the  factor  y  as  recommended  by  Bates  are  those 
worked  out  by  Lewis  for  the  15-degree  involute  teeth.  For  com- 
mercial pitches  and  corresponding  gear  faces,  see  Table  80. 

(6)  Formula  for  Wuest  gears. — In  a  comprehensive  paper 
before  the  American  Society  of  Mechanical  Engineers,  Mr.  P. 
C.  Day  of  The  Falk  Co.  gave  a  simple  formula  for  determining  the 
safe  working  load  on  the  teeth  of  Wuest  helical  gears.  The 
formula  is  empirical,  as  it  is  based  upon  the  results  obtained  from 
several  years  of  experience  with  such  gears.  Using  as  far  as 
possible  the  notation  given  in  the  preceding  discussion,  the  safe 
working  load  is  as  follows: 


W  =  0.4  Sp'f, 


(388) 


in  which  the  factor  S  represents  the  shearing  stress  on  a  section 
taken  at  the  pitch  line.     This  shearing  stress  varies  with  the 


ART.  258] 


MATERIALS  FOR  HELICAL  GEARING 


357 


pitch-line  speed,  as  shown  in  Fig.  176.  The  length  of  the  total 
face  of  the  gear  should  be  at  least  five  times  the  circular  pitch, 
and  for  average  conditions  six  times  the  pitch  gives  satisfactory 
service.  When  the  gear  ratio  is  high,  the  face  may  be  made  ten 
times  the  circular  pitch,  provided  the  pinion  and  gear  are  mounted 
on  rigid  bearings  located  close  together. 

When  the  load  transmitted  by  the  gears  fluctuates  from  a  mini- 
mum to  a  maximum,  as  in  the  case  of  single-acting  pumps  and 
mine  hoists,  the  gears  should  be  designed  for  a  load  which  repre- 
sents an  average  between  the  maximum  and  mean  loads.  The 
gears  used  in  connection  with  motor-driven  machine  tools  should 


500 


zooo 


2500 


1000  1500 

Velocity    in    ft.  per    min. 

FIG.  176. 

be  designed  to  transmit  a  load  equivalent  to  the  rated  output  of 
the  motor  at  a  speed  which  is  taken  as  the  mean  between  the 
maximum  and  minimum  revolutions  per  minute.  The  design  of 
high-ratio  and  rolling-mill  transmissions  must  receive  special 
consideration,  and  should  be  left  to  the  engineers  of  the  company 
that  manufacture  such  gears. 

258.  Materials  for  Helical  Gearing. — In  general,  soft  mate- 
rials such  as  rawhide,  fiber,  and  cloth  should  never  be  used  for  the 
pinion.  Some  manufacturers  do  not  consider  it  good  practice  in 
high-ratio  transmissions  to  use  cast  iron  for  cut  double-helical 
pinions,  claiming  that  a  forged-steel  pinion  will  cost  but  little 
more,  and,  due  to  its  better  wearing  qualities,  will  give  increased 
life  to  the  transmission.  When  the  tooth  pressures  are  moderate, 


358 


HELICAL  GEAR  CONSTRUCTION 


[CHAP.  XI 


cast  iron  or  semi-steel  is  preferred  to  steel  casting  for  gears  of 
large  diameter;  but  when  the  loads  are  heavy,  steel  casting  is 
generally  more  economical.  The  carbon  content  of  the  grade  of 
steel  casting  used  ordinarily  for  gears  varies  from  0.25  to  0.30 
per  cent.  When  the  gear  and  pinion  are  both  made  of  steel,  the 
best  results  are  obtained  by  making  the  pinion  of  a  different  grade 
of  steel  than  that  used  for  the  gear;  for  example,  with  a  gear  made 
of  steel  casting  having  a  carbon  content  of  0.25  to  0.30  per  cent., 
the  pinion  should  be  made  of  a  0.40  to  0.50  per  cent,  carbon-steel 
forging.  For  high-pitch  line  velocities,  alloy-steel  pinions  sub- 
jected to  a  heat  treatment  are  recommended.  Frequently  the 
pinion  teeth  are  cut  integral  with  the  shaft. 


FIG.   177. 

259.  Double -helical  Gear  Construction. — (a)  Rim. — For  large 
gears,  The  Falk  Co.  has  found  that  whenever  possible  the  rim 
should  be  made  solid,  and  when  the  diameter  of  the  gear  exceeds 
7  feet  the  hub  should  be  split.  The  split  in  the  hub  should  be 
placed  midway  between  two  arms;  thus  when  six  arms  are  used, 
as  is  their  usual  practice,  two  of  these  arms  are  perpendicular  to 
the  split.  The  Falk  Co.  has  found  that  with  this  arrangement  the 
casting  will  contract  very  evenly,  so  that  the  rough  gear  blank  on 
leaving  the  sand  is  practically  round.  It  is  claimed  that  such  a 
construction,  when  used  with  eight  arms,  produces  a  casting  that 


ART.  259] 


HELICAL  GEAR  CONSTRUCTION 


359 


is  distorted.     Figs.  177  and  178  show  two  large  gears  made  of 
steel  casting  and  built  by  The  Falk  Co. 


FIG.   178. 


FIG.   179. 


Large  double-helical  gears  transmitting  heavy  loads  are  fre- 
quently made  with  a  steel-casting  rim,  cast  in  halves  and  bolted  to 
a  cast-iron  spider.  The  rims  of  such  gears  are  shown  in  Figs. 


360 


HELICAL  GEAR  CONSTRUCTION 


[CHAP.  XIV 


179  and  180,  and  the  cast-iron  spider  for  the  latter  is  shown  in 
Fig.  181.  In  order  to  relieve  the  coupling  bolts  between  the  rim 
and  the  spider  of  all  shearing  action,  large  heavy  keys  are  fitted 


FIG.  180. 


FIG.   181. 


between  the  rim  and  the  arms  of  the  spider.  The  rim,  being  made 
in  halves,  has  the  joints  split  parallel  to  the  tooth  angle.  These 
rim  joints  should  always  be  located  between  two  teeth  as  shown 


ART.  259] 


HELICAL  GEAR  CONSTRUCTION 


361 


in  Fig.  182.  Joints  made  in  this  manner  do  not  weaken  the  teeth, 
nor  do  they  interfere  with  the  smooth  operation  of  the  gear. 
Bolts  and  shrink  links  as  shown  in  Fig.  182  are  used  for  fastening 
together  the  two  halves  of  the  rim. 

Another  design  of  a  rim  joint  is  shown  in  Fig.  183,  and  as  in 
the  design  just  described,  the  steel-casting  rim  is  fastened  to  a 
cast-iron  spider  by  means  of  bolts  and  shrink  links.  This  joint, 
however,  differs  from  the  one  shown  in  Fig.  182  in  that  a  tongue 
and  groove  are  used,  the  tendency  of  which  is  to  weaken  the  tooth 
along  the  joint,  as  is  evident  from  an  inspection  of  Fig.  183. 


FIG.   182. 

In  Fig.  184  is  shown  an  excellent  design  of  a  heavy  steel  casting 
double-helical  gear,  cast  in  halves.  The  joint  is  made  through 
the  arms,  and  a  series  of  studs  as  shown  hold  the  two  halves  of 
the  gear  together.  The  studs  in  the  arms  are  fitted  accurately 
into  reamed  holes,  while  those  in  the'  hub  and  under  the  rim  are 
fitted  very  loosely,  because  it  is  impossible  to  ream  these  holes. 
The  split  in  the  rim  is  made  between  two  teeth  and  parallel  to 
the  teeth. 

The  rim  sections  in  common  use  are  illustrated  in  the  various 


362 


HELICAL  GEAR  CONSTRUCTION          [CHAP.  XIV 


r— 28' 
I      I 


Spot  Face 


3- 


3H  T- 


U-27' 


FIG.  183. 


FIG.  184. 


ART.  259] 


HELICAL  GEAR  CONSTRUCTION 


363 


figures  mentioned  in  the  preceding  discussion.  According  to 
Bates,  the  finished  rim  thickness  under  the  teeth  of  cut  double- 
helical  gears  may  be  arrived  at  by  the  following  empirical 
formula : 

2       1" 
Rim  thickness  =  -  +  -  (389) 

In  Fig.  185  is  shown  a  double-herringbone  pinion,  the  teeth  of 
which  are  cut  integral  with  the  shaft.  This  shaft  with  the  double 
pinion  is  used  for  driving  two  large  gears  of  a  rolling-mill  drive. 

(6)  Arms. — Arms  of  elliptical  cross-section  should  never  be 
used  for  double-helical  gearing  for  the  reason  that  they  lack  rigid- 
ity at  right  angles  to  the  direction  of  rotation.  For  gears  not 
exceeding  40  inches  in  diameter,  and  having  a  length  of  face 


FIG.  185. 

approximating  one-sixth  to  one-eighth  of  the  diameter,  Bates 
recommends  the  use  of  cross-shaped  arms.  With  gears 
having  wider  faces  than  those  just  mentioned,  the  H-section 
similar  to  those  shown  in  Figs.  178  and  181  should  be  used. 
Furthermore,  according  to  the  same  authority,  the  face  of  cut 
gears  should  never  be  made  less  than  one-tenth  of  the  pitch 
diameter,  if  the  gear  is  to  possess  sidewise  rigidity  and  no  vibra- 
tion is  to  be  set  up  in  the  transmission.  For  heavy  rolling-mill 
drives,  the  face  of  the  gears  is  unusually  long,  and  for  such  gears 
The  Falk  Co.  recommends  the  use  of  double  arms  of  U  cross-sec- 
tion. In  general,  the  section  of  the  arms  should  be  made  con- 
siderably heavier  where  they  join  the  hub  so  as  to  insure  sound 
castings. 

260.  Mounting  of  Double -helical  Gears.— Due  to  the  high 
speeds  at  which  double-helical  gears  are  used,  the  frames  and 


364  CIRCULAR  HERRINGBONE  GEARS         [CHAP.  XIV 

bearings  supporting  such  gears  must  be  made  heavy  and  rigid. 
The  shafts  must  all  be  in  true  alignment,  and  the  pinion  and  gear 
must  have  the  supporting  bearings  located  close  up  to  the  hubs. 
The  gear  with  its  mating  pinion  should  be  aligned  correctly  so  as 
to  eliminate  all  end  thrust.  Means  for  lubricating  the  trans- 
mission must  be  provided,  and  the  whole  arrangement  should  be 
made  accessible  for  inspection.  For  a  high-ratio  transmission 
running  at  a  high  rotative  speed,  the  pinion  is  generally  integral 
with  its  shaft,  and  the  latter  is  driven  by  the  prime  mover  or  mo- 
tor through  the  medium  of  a  flexible  coupling. 

261.  Circular  Herringbone  Gears. — Several  years  ago,  the  R. 
D.  Nuttall  Co.  developed  and  introduced  a  new  form  of  generated 
tooth  gear  to  which  the  term  circular  herringbone  was  applied. 
Such  a  gear  has  continuous  teeth  extending  across  its  face  in  the 
form  of  circular  arcs.  The  teeth  are  generated  by  two  cutters, 
one  for  each  side  of  the  tooth.  The  profile  of  these  cutters  is  an 
involute  rack  tooth,  and  the  pressure  angle  for  the  middle  section 
of  the  gear  tooth  is  20  degrees.  This  angle,  however,  varies 
slightly  for  all  the  other  sections  of  the  tooth,  increasing  as  the 
sections  approach  the  end  of  the  gear  face.  The  Nuttall  Co.  has 
adopted  as  a  standard  for  these  gears  a  short  tooth  having  the 
following  proportions: 

1.  The  tooth  profile  is  made  a  20-degree  involute. 

2.  The  length  of  the  addendum  is  made  0.25  p' '. 

3.  The  clearance  is  made  0.05  p'. 

4.  The  whole  depth  of  the  tooth  is  made  0.55  p'. 

5.  The  radius  of  curvature  of  the  tooth  and  that  of  the  face  of 
the  gear  are  made  equal,  and  should  never  be  less  than  twenty- 
four  divided  by  the  diametral  pitch. 

According  to  the  manufacturers,  the  circular  herringbone  gears 
have  all  the  advantages  of  double-helical  gears,  and  in  addition 
two  special  advantages  are  claimed. 

1.  Due  to  the  fact  that  the  tooth  is  continuous  and  not  grooved 
at  the  center,  it  is  stronger  and  at  the  same  time  the  rim  is  re- 
inforced. 

2.  The  lubrication  is  applied  more  readily,  since  the  curved 
tooth  acts  like  a  cup. 

WORM  GEARING 

The  type  of  screw  gearing  commonly  called  worm  gearing  is 
used  for  transmitting  power  and  obtaining  high  speed  reductions 


ART.  263]  HINDLEY  WORM  GEARING  365 

between  non-intersecting  shafts  making  an  angle  of  90  degrees 
with  each  other.  There  are  two  classes  of  worm  gearing  in 
common  use,  each  of  which  possesses  certain  advantages  over 
the  other. 

262.  Straight  Worm  Gearing. — The   class   of  worm  gearing 
most  frequently  used  is  that  in  which  the  worm  is  straight  or  of  a 
cylindrical  shape.     The  threads  of  such  a  worm  have  an  axial 
pitch  that  is  constant  for  all  points  between  the  top  and  the  root 
of  the  threads.     Strictly  speaking,  there  are  two  types  of  straight 
worm  gearing.     In  the  first  of  these  types,  generally  called  the 
ordinary  worm  and  gear,  the  hob  used  for  machining  the  worm 
gear  is  of  constant  diameter  and  is  fed  radially  to  the  proper 
depth  into  the  blank,  both  hob  and  blank  being  rotated  in  correct 
relation  to  each  other.     The  teeth  produced  are  not  theoretically 
correct  in  shape.     In  place  of  a  cylindrical  hob,  one  that  tapers 
may  be  used,  and  by  feeding  it  into  the  gear  blank  longitudinally 
at  right  angles  to  the  axis  of  the  blank  instead  of  radially  as  in  the 
preceding  case,  the  worm  gear  produced  has  teeth  that  approach 
very  closely  the  theoretical  form.     Gears  cut  by  the  latter  method 
have  given  much  better  service  and  higher  efficiencies  than  similar 
gears  cut  by  the  first  method. 

Due  to  the  higher  grade  of  product  obtained  by  the  use  of  a 
taper  hob,  the  second  type  of  worm  and  gear  is  employed  to  a 
considerable  extent  in  the  rear  axle  drives  of  auto-trucks  and  mo- 
tor cars.  The  efficiency  and  load-carrying  capacity  are  practi- 
cally the  same  as  for  the  hollow-worm  type  of  gearing  described 
in  the  following  article. 

263.  Hindley  Worm  Gearing. — In  the  second  class  of  worm 
gearing,  the  worm  has  a  shape  similar  to  that  of  an  hour  glass. 
It  was  introduced  by  Hindley  in  connection  with  his  dividing 
engine,  and  worms  having  a  hollow  face  are  generally  called 
Hindley  worms.     As  may  be  seen  from  Fig.  186,  the  worm  is  made 
smaller  in  the  center  than  at  its  ends,  so  that  it  will  conform  to 
the  shape  of  the  gear.     Since  there  is  a  larger  contact  surface 
between  the  mating  teeth  than  in  the  straight  worm  class,  the 
wear  is  reduced  and  it  is  possible  to  use  a  smaller  pitch  and  face 
of  gear  for  a  given  transmission.     In  the  Hindley  worm  the  axial 
pitch  varies  at  every  point,  since  the  angle  of  the  helix  changes 
constantly  throughout  the  length  of  the  worm.     At  the  center 
of  the  worm,  the  helix  angle  is  much  greater  than  at  the  ends, 
as  is  evident  from  an  inspection  of  Fig.  186. 


366 


HINDLEY  WORM  GEARING 


[CHAP.  XIV 


Hindley  worm  gearing  is  produced  by  the  bobbing  process, 
but  since  the  shape  of  the  worm  is  made  to  conform  to  the  cir- 
cumference of  the  gear,  it  follows  that  such  worms  are  not 
interchangeable.  In  other  words,  a  worm  intended  for  a  gear 
containing  36  teeth  of  a  given  pitch  will  not  mesh  correctly 
with  a  gear  having  54  teeth  of  the  same  pitch.  In  order  to  ob- 
tain good  results  with  the  use  of  Hindley  worm  gearing,  the 
following  requirements  must  be  met: 

1.  The  center  .distance  between  the  worm  and  gear  must  be 
exact. 


FIG.   186. 

2.  The   center  of  the  worm  must  conform  exactly  with  the 
center  of  the  gear  so  as  to  avoid  any  longitudinal  displacement  of 
the  worm. 

3.  The  worm  axis  must  be  in  proper  alignment,  relative  to  the 
gear. 

Experiments  conducted  on  well-designed  and  properly  mounted 
worm  gears,  as  used  in  motor-car  work,  show  that  the  efficiency 
and  load-carrying  capacity  of  the  hollow  worm  are  slightly  greater 
than  those  obtained  by  means  of  the  straight  worm,  although  the 
difference  is  small. 

264.  Materials  for  Worm  Gearing. — In  general,  a  worm  gear 
transmission  gives  satisfactory  service  when  the  worm  is  made  of 
a  low-carbon  steel  and  the  gear  of  a  good  grade  of  bronze.  The 


ART.  264] 


MATERIALS  FOR  WORM  GEARING 


367 


steel  for  the  worm  should  have  a  carbon  content  that  will  permit 
of  heat-treatment  without  producing  serious  distortion  of  the 
worm.  The  heat-treatment  that  is  generally  used  is  one  of 
carbonizing  or  case-hardening.  For  this  purpose  some  manu- 
facturers prefer  a  nickel  steel  with  a  low  carbon  content,  while 
others  specify  an  open-hearth  high-carbon  steel.  In  Table  82 
are  given  six  different  gear  bronzes  that  the  Wm.  Cramp  and 
Sons  Ship  and  Engine  Building  Co.  has  found  to  be  satisfactory 
for  the  various  classes  of  service  indicated. 

TABLE  82. — CRAMP'S  GEAR  BRONZES 


Bronze 
No. 

Tensile 
strength 

Elastic 
limit 

Wt. 
per 
cu.  in. 

Range  of 
load 

Permis- 
sible 
r.p.m.  of 
worm 

Class  of  service 

1 

40,000 

20,000 

0.316 

not  over  1,500 

1,500 

Light  loads  and  high  speeds. 

2 

40,000 

20,000 

0.319 

3,000  to    4,000 

1,000 

Moderate  loads  and  speeds. 

3 

45,000 

22,000 

0.321 

3,000  to    4,000 

1,000 

Moderate   loads   and   speeds 

when  excessive  wear  is  ex- 

pected. 

4 

30,000 

15,000 

0.300 

5,000  to  25,000 

200  to  400 

For      continuous      moderate 

loads       with       intermittent 

heavy  load. 

5 

35,000 

18,000 

0.325 

3,000 

200 

For  average  running  condi- 

1,000 to    1,500 

600  to  900 

tions  of  light  loads  and  mod- 

erate    speeds     with     heavy 

starting  torque. 

Parson's 

65,000 

30,000 

0.305 

10,000  to  50,000 

200 

For    heavy    loads    and    slow 

Man. 

speeds  under  excessive  strain 

Bronze. 

and  shock. 

From  the  preceding  statements  it  should  not  be  understood 
that  steel  and  bronze  are  the  only  materials  that  are  satisfactory 
for  worm  gearing.  A  carbonized  steel  worm  and  a  gear  made  of 
a  high-grade  semi-steel  casting  will  give  good  service  for  moderate 
loads  and  speeds.  For  light  loads  and  low  speeds,  a  carbonized- 
steel  worm  with  a  gear  made  of  close-grained  cast  iron  will  prove 
satisfactory. 

265.  Tooth  Forms.— (a)  Straight  worm— The  standard  form 
of  tooth  used  for  the  ordinary  worm  gearing  is  that  proposed 
and  adopted  as  a  standard  by  the  Brown  and  Sharpe  Mfg.  Co. 
As  shown  in  Fig.  187,  the  sides  of  the  worm  thread  make  an  angle 
of  29  degrees  with  each  other,  or  in  other  words,  the  pressure 
angle  is  14>^  degrees.  This  form  of  worm  thread  is  produced 
by  a  straight-sided  tool  having  flat  ends,  and  for  the  various 
pitches  in  use,  the  proportions  .may  be  taken  from  Table  83. 


368 


TOOTH  FORMS  FOR  WORM  GEARING      [CHAP.  XIV 


TABLE  83. — STANDARD  29°  WORM  THREADS 


The  teeth  on  the  gear  which  mesh  with  a  worm  having  teeth 
according  to  the  proportions  shown  in  Table  83  are  given  an 

involute  form,  and, 
according  to  the 
Brown  and  Sharpe 
Mfg.  Co.,  such  gears 
should  always  have 
more  than  31  teeth 
in  order  to  avoid 
undercutting  of  the 
teeth. 

In  modern  manu- 
facturing, the  so- 
called  straight 
worms  are  no  longer 
turned  on  a  lathe, 
but  are  milled. 
With  the  use  of  the 
29-degree  thread, 

there  is  some  difficulty  in  milling  such  a  worm  when  the  helix 
angle   approaches  28   degrees.     To  obviate  any   difficulty  that 


Circular 
pitch 

Threads 
per 
inch 

Tooth 
height 
above 
pitch  line 

Total 
height 
of  tooth 

Width  of  tooth  at 

Top 

Bottom 

X 

4 

0.0796 

0.1716 

0.0838 

0.0775 

Yi 

3M 

0.0909 

0.1962 

0.0957 

0.0886 

y* 

3 

0.1061 

0.2288 

0.1117 

0.1033 

2A 

2*A 

0.1273 

0.2746 

0.1340 

0.1240 

1A 

2 

0.1592 

0.3433 

0.1675 

0.1550 

X 

IX 

0.2122 

0.4577 

0.2233 

0.2066 

3A 

IH 

0.2387 

0.5150 

0.2512 

0.2325 

i 

i 

0.3183 

0.6866 

0.3350 

0.3100 

IK 

X 

0.3979 

0.8583 

0.4187 

0.3875 

'IX 

X 

0.4775 

1.0299 

0.5025 

0.4650 

IX 

K 

0  .  5570 

1.2016 

0.5862 

0.5425 

2 

y* 

0.6366 

1  .  3732 

0.6708 

0.62'00 

FIG.   187. 


may  arise,  the  angle  between  the  sides  of  the  tooth  is  made  larger 
than  29  degrees.  Some  designers  have  adopted  an  angle  of  60 
degrees,  while  others  vary  the  angle  for  different  helix  angles. 


ART.  266]          LOAD  CAPACITY  OF  WORM  GEARING  369 

(6)  Hindley  worm. — According  to  the  practice  of  the  Keystone- 
Hindley  Gear  Co.,  the  angle  included  between  the  sides  of  the 
teeth  varies  considerably,  as  is  shown  by  the  following: 

1.  For  single-threaded  worms,  the  angle  is  made  29  degrees. 

2.  For  double-threaded  worms,  the  angle  is  made  35  degrees. 

3.  For  triple-threaded  worms,  the  angle  is  made  35  degrees. 

4.  For  quadruple-threaded  worms,  the  angle  is  made  37^ 
degrees. 

5.  For  worms  of  small  diameter  having  from  two  to  four 
threads,  the  tooth  angle  is  made  as  high  as  52  degrees. 

Furthermore,  this  same  company  has  no  uniform  depth  of 
tooth,  as  it  varies  from  75  to  100  per  cent,  of  the  circular  pitch, 
with  an  average  of  about  85  per  cent. 

In  the  Lanchester  worm  gearing,  which  is  probably  one  of  the 
most  efficient  types  of  Hindley  gearing  in  use,  the  side  of  the  tooth 
is  given  a  slope  of  1  in  2. 

266.  Load  Capacity. — The  permissible  load  upon  the  worm- 
gear  teeth  depends  more  upon  the  heating  effect  and  wear  pro- 
duced than  upon  the  strength  of  the  teeth.  If  the  oil  film  be- 
tween the  teeth  in  contact  breaks  down,  due  to  high  pressure 
or  to  thinning  of  the  lubricant  caused  by  high  temperatures, 
excessive  heating  and  wear  will  result.  If  not  remedied,  this 
will  in  a  short  time  destroy  the  gear  or  worm,  or  both.  The 
formulas  in  use  for  determining  the  permissible  load  on  worm- 
gear  teeth  are  all  of  an  empirical  nature,  having  the  following 
form: 

W  =  Cfp',  (390) 

in  which  /  and  p'  denote  the  face  and  circular  pitch,  respectively, 
and  C  is  a  coefficient  depending  upon  the  speed,  pressure,  and 
temperature.  This  coefficient  must  be  determined  by  means  of 
experiments. 

In  1902,  Prof.  C.  Bach  and  E.  Roser  made  an  experimental 
investigation  of  a  triple-threaded  soft-steel  worm  and  bronze 
worm  gear  running  under  various  conditions.  The  pitch  diam- 
eter of  the  worm  was  a  trifle  over  3  inches  and  the  lead  was  3 
inches,  thus  giving  a  helix  angle  of  17  degrees  34  minutes. 
The  worm  gear  contained  30  teeth  of  involute  profile  having  a 
pressure  angle  of  14J^  degrees.  The  results  of  these  tests  were 
published  in  the  Zeitschrift  des  Vereins  deutscher  Ingenieure 
of  Feb.  14,  1903,  also  in  the  American  Machinist,  July  16  and  23, 


370  STRENGTH  OF  WORM-GEAR  TEETH       [CHAP.  XIV 

1903.  The  expression  for  the  allowable  load  on  the  worm  drive 
as  proposed  by  Bach  and  Roser  is  more  or  less  involved,  and 
since  it  is  based  upon  the  investigation  of  a  single  worm  trans- 
mission, its  adoption  as  a  working  formula  may  be  questioned. 
The  Bach  and  Roser  formula,  assuming  continuous  service,  is 
as  follows: 

W  =  (mt  +  n)/V,  (391) 

in  which  /'  denotes  the  face  of  the  worm  gear  measured  in  inches 
on  an  arc  at  the  base  of  the  teeth;  pr  denotes  the  divided  pitch 
of  the  worm  or  the  circular  pitch  of  the  worm  gear;  t  denotes  the 
rise  in  degrees  F.  in  the  temperature  of  the  oil  in  the  reservoir; 
m  and  n  are  experimental  coefficients  depending  upon  the  velocity 
of  the  teeth.  The  relations  existing  between  the  velocity  V 
in  feet  per  minute  and  the  coefficients  m  and  n  are  given  by  the 
following  expressions  : 


-  356 

For  ordinary  working  conditions,  the  temperature  rise  t  in 
(391)  may  be  assumed  to  vary  from  80°  to  100°F.  If  the  drive 
is  to  be  installed  in  a  place  where  the  prevailing  temperature  is 
high,  the  magnitude  of  t  should  be  based  upon  the  temperature 
at  which  the  lubricant  used  in  the  drive  loses  its  lubricating 
qualities.  In  view  of  the  fact  that  formula  (391)  is  based  upon 
continuous  service,  it  seems  reasonable  that  for  intermittent 
service  the  permissible  load  as  determined  by  (391)  may  be  in- 
creased; in  other  words,  instead  of  designing  the  drive  for  the 
maximum  load,  the  average  load  might  be  used  in  arriving  at  the 
safe  dimensions  of  the  worm-gear  teeth. 

267.  Strength  of  Worm-gear  Teeth.  —  It  may  occasionally  be 
necessary  to  investigate  the  teeth  of  the  worm  gear  for  strength, 
and  in  such  cases  the  formulas  derived  for  spur  gearing  may  be 
used  by  making  the  following  modifications: 

(a)  For  cast  gearing,  the  load  W  should  be  considered  as  coming 
upon  a  single  tooth. 

(b)  For  cut  gearing,  assume  the  load  W  as  equally  distributed 
among  all  the  teeth  in  actual  contact  as  given  by  (408). 

(c)  For  the  magnitude  of  /in  the  spur-gear  formula,  determine 
the  actual  length  of  the  gear  tooth  at  the  base  of  the  tooth, 


ART.  268]         FORCE  ANALYSIS  OF  WORM  GEARING 


371 


268.  Force  Analysis  of  Worm  Gearing. — In  order  to  arrive 
at  the  probable  pressure  coming  upon  the  various  bearings  used 
in  the  mounting  of  a  worm-gear  drive,  it  is  necessary  to  deter- 
mine the  relation  existing  between  the  turning  force  on  the  worm 
and  the  tangential  resistance  on  the  worm  gear.  Having 
established  this  relation,  the  magnitudes  of  the  various  com- 
ponents of  the  tangential  resistance  may  then  be  determined, 
and  from  these  components  the  pressures  upon  the  bearings 
may  be  found. 

(a)  Relation  between  effort  and  load. — The  relation  between  the 
equivalent  turning  force  P  on  the  worm  and  the  tangential  load 
W  upon  the  worm  gear  may  be  obtained  as  follows: 


FIG.   188. 

Referring  to  Fig.  188,  the  vector  N  represents  the  normal 
reaction  between  the  teeth  at  the  point  of  contact  0.  The  symbol 
r  denotes  the  pitch  radius  of  the  worm;  a  the  angle  of  the  helix 
of  the  worm;  /3  the  pressure  angle  or  the  angle  the  side  of  the 
thread  makes  with  a  line  at  right  angles  to  the  center  line  of  the 
worm.  The  angle  <p  is  the  angle  of  friction  for  the  materials  in 
contact. 

Disregarding  the  frictional  resistances,  the  components  of  the 
normal  force  N  along  the  X,  Y,  and  Z  axes  are,  respectively, 

Nc  =  NCOS  6  COS  a. 

Ny  =  NCOS  6  sin  a 
N,  =  Nsm  0 

Assuming  that  the  worm  shown  in  Fig.  188  rotates  in  the  direc- 
tion as  indicated  in  Fig.  188(a),  the  force  pN  due  to  friction 


372  FORCE  ANALYSIS  OF  WORM  GEARING    [CHAP.  XIV 

upon  the  worm  acts  along  the  tangent  to  the  helix.  This  force 
of  friction  tends  to  increase  or  decrease  the  components  found 
above;  hence  resolving  /JV  along  the  X  and  Y  axes,  we  obtain 
the  remaining  components: 


a 


N'x  = 

N'y  —  pcos  a 

Each  of  the  five  components  is  shown  in  Fig.  188.  Now 
adding  the  components  along  the  same  lines  of  action,  we  obtain 
the  following  expressions: 

The  magnitude  of  the  tangential  force  exerted  by  the^worm 
gear  upon  the  worm  teeth  is 

W  =  Nx  -  N'x  =  N  (cos  6  cos  a  -  /*  sin  «)  (393) 

The  magnitude  of  the  turning  force  P  required  at  the  pitch 
radius  of  the  worm  is  obtained  by  adding  Ny  and  N'v  thus 

P  =  Nv  +  N'y  =  N  (cos  B  sin  a  +  /*  cos  a)  (394) 

The  force  S,  causing  a  downward  pressure  upon  the  worm  shaft 
or  an  upward  pressure  upon  the  worm-gear  shaft,  has  a  magni- 
tude given  by  Nz  above,  namely, 

S  =  Ne  =  N  sin  0  (395) 

The  relation  between  P  and  W  may  now  be  obtained  by  com- 
bining (393)  and  (394);  thus 

P  = 


cos  6  cos  a  —  ju  sin  a 

Denoting  the  ratio  of  ju  to  cos  6  by  tan  <pr,  (396)  reduces  to  a  sim- 
ple form  of  expression  which  is  similar  to  that  derived  for  screws, 
namely, 

P  =  W  tan(«  +  *>')  (397) 

Letting  p'  denote  the  lead  of  the  worm  and  writing  //  =  tan  ^', 
(397)  may  be  put  into  the  following  form: 


From  Fig.  188,  it  is  evident  that 
tan  6  =  tan  ]8  cos  a 


Hence  /*'•'•?=  -£-  =  M  Vl  +  cos2  a  tan2  0  (399) 

cos  0 


ART.  269] 


BEARING  PRESSURES 


373 


Now  //  may  be  considered  a  "new  coefficient  of  friction"  peculiar 
to  worm  gearing  and  its  magnitude  may  be  obtained  by  means 
of  (399). 

Combining  (393)  and  (395)  and  reducing  to  a  simple  form,  the 
magnitude  of  the  force  S  in  terms  of  W  is  given  by  the  following 
expression : 

S  =  W  h-^M  (400) 

LI  —  ju  tan  a  A 

(b)  Efficiency  of  worm  gearing. — An  expression  for  the  efficiency 
of  a  worm  and  gear  may  now  be  determined.  In  the  ideal  trans- 
mission, namely,  one  having  all  of  the  frictional  resistances  elimi- 
nated, it  is  apparent  that  the  effort  PQ  required  at  the  pitch 
radius  of  the  worm  is  as  follows: 


Po  =  W  tan  a 


(401) 


Hence  the  efficiency  of  the  worm  and  gear,  not  taking  into  consid- 
eration the  frictional  resistances  of  any  of  the  bearings  used  in 
the  mounting,  is  given  by  the  following  formula: 


_         _ 

17  ~          ~ 


tan 


( 


269.  Bearing  Pressures.  —  (a)  Worm  shaft.  —  The  worm  shaft 
is  generally  supported  on  two  bearings,  each  of  which  must  be 


FIG.   189. 

capable  of  withstanding  the  pressure  coming  upon  it  due  to  the 
forces  P,  W,  and  S.  In  addition  to  the  transverse  forces,  the 
worm  shaft  is  also  subjected  to  a  thrust,  and  for  that  reason 
a  thrust  bearing  must  be  provided.  When  ball  bearings  are  used 
for  mounting  the  worm  shaft,  it  is  possible  to  select  a  type  of 
radial  bearing  that  is  capable  of  taking  care  of  a  certain  amount 
of  end  thrust  in  addition  to  the  transverse  load.  Such  a  bearing 
makes  the  installation  of  a  special  thrust  bearing  unnecessary. 
In  Fig.  189  is  shown  a  worm  shaft  mounted  on  radial  bearings 


374 


BEARING  PRESSURES 


[CHAP.  XIV 


that  are  capable  of  taking  an  end  thrust  equivalent  to  one  and 
one-half  times  the  radial  load.  Assuming  that  the  turning  force 
P  and  the  downward  pressure  S  are  applied  midway  between 
the  bearings  A  and  B,  each  of  these  bearings  is  subjected  to  a 
pressure  equal  to  one-half  of  these  forces.  Since  S  is  at  right 

angles  to  P,  the  compo- 
nents of  these  forces  at 
the  bearings  are  at  right 
angles  to  each  other.  The 
tangential  force  W,  in  ad- 
dition to  causing  an  end 
thrust  upon  the  bearing  A, 
also  produces  a  pressure 

equal    to    —j-  upon     each 

bearing,  the  one  at  A  act- 
ing downward  and  that  at 
B  upward.  Hence  the  re- 
sultant pressure  upon  the 
bearing  A  is  as  follows : 


A  = 


-y;  (403) 


and  the  resultant  pressure 
upon  the  bearing  B  is 


(404) 


ir         r 

_T  "  2] 


Having  determined  the  magnitudes  of  the  bearing  pressures  and 
thrusts,  the  size  of  bearing  may  now  be  selected  from  tables 
furnished  by  the  manufacturers  of  such  bearings. 

(6)  Worm-gear  shaft. — The  pressure  exerted  upon  the  bearings 
supporting  the  worm  gear  depend  upon  the  magnitudes  of  the 
forces  P,  Wt  and  S}  as  well  as  upon  the  method  of  mounting  the 
gear.  The  transmission  illustrated  in  Fig.  190  has  the  gear  sup- 
ported on  ball  bearings  mounted  on  the  extended  hubs  of  the 
gear.  In  some  installations,  the  gear  is  keyed  to  a  shaft  which 
in  turn  is  supported  on  proper  bearings.  If  the  bearings  C  and 
D  in  Fig.  190  are  located  symmetrically  with  respect  to  the  center 
plane  of  the  worm  gear,  the  pressures  upon  them  due  to  the  forces 


ART.  270]  WORM  AND  GEAR  CONSTRUCTION  375 

S  and  W  will  be  equal  to  one-half  of  these  forces.  The  force  P 
tends  to  move  the  gear  along  its  axis,  thus  producing  a  thrust  on 
the  bearing  D,  and  at  the  same  time  this  force  introduces  a  trans- 
verse pressure  upon  both  of  the  bearings.  The  transverse  pres- 

DD 

sures  due  to  P  have  a  magnitude  - — >  the  one  acting  upward 

c 

on  the  bearing  C  and  the  other  downward  on  the  bearing  D. 
Since  P  causes  an  end  thrust,  it  is  necessary  that  the  radial  ball 
bearings  used  for  supporting  the  gear  be  of  a  type  that  is  capable 
of  supporting  a  thrust  in  addition  to  the  radial  load.  Proceeding 
as  in  the  case  of  the  worm  shaft,  the  following  expressions  are 
obtained : 
The  resultant  radial  load  on  the  bearing  C  is 


(405) 
and  the  resultant  radial  load  on  the  bearing  D  is 

(406) 


270.  Worm  and  Gear  Construction. — In  many  worm  gear 
transmissions,  the  worm  is  made  integral  with  the  shaft  as  shown 
in  Figp.  186  and  194  to  197,  inclusive.  However,  occasionally  in 
machine  tools  using  worm  drives,  it  is  desirable  to  make  the  worm 
separate  from  the  shaft  and  fasten  it  to  the  latter  by  means  of 
keys  or  taper  pins  as  shown  in  Figs.  187  and  191. 

Worm  gears  made  of  cast  iron,  semi-steel,  or  steel  casting  are 
constructed  in  the  same  way  as  ordinary  spur  or  helical  gearing. 
If  the  gear  is  relatively  small  the  solid  or  web  construction  shown 
in  Fig.  191  is  used.  With  gears  of  large  diameter  considerable 
material  may  be  saved  by  the  use  of  arms  in  place  of  a  web. 
The  dimensions  of  the  arms  may  be  determined  by  the  formulas 
given  in  Art.  229.  When  bronze  is  used  for  the  gear  the  cost 
may  be  kept  down  by  making  the  rim  of  bronze,  as  shown  in 
Fig.  186,  and  bolting  it  to  a  spider  made  of  cast  iron,  semi- 
steel,  or  steel  casting.  An  example  of  a  worm  gear  having  a 
bronze  rim  bolted  to  a  cast-iron  spider  is  shown  in  Fig.  197. 

(a)  Length  of  worm. — In  the  worm  and  gear,  shown  diagram- 
matically  in  Fig.  192,  the  symbol  D  denotes  the  pitch  diameter  of 
the  gear,  and  a  the  addendum  of  the  teeth.  The  intersections 
of  the  addendum  line  of  the  worm  with  the  addendum  circle 


376 


WORM  AND  GEAR  CONSTRUCTION       [CHAP.  XIV 


of  the  gear  are  the  extreme  points  of  available  tooth  contact; 
thus  the  chord  AB  represents  the  minimum  length  of  the  straight 


FIG.   191. 

type    of  worm   in    order   that    complete  tooth  action  may  be 
obtained.     The  expression  for  the  length  AB  is  as  follows : 


A  = 


"20JD  =  (D  +  2  a)  sin  0 


(407) 


For  worms  of  the  Hindley  type, 
the  length  as  recommended  by 
Lanchester  is  such  that  the  differ- 
ence between  the  maximum  and 
minimum  diameters  is  approxi- 
mately 7  to  8  per  cent,  of  the  latter. 

Having  determined  the  length 
of  the  chord  AB  by  means  of  (407), 
the  number  of  gear  teeth  in  actual 
contact  with  the  worm  is  then 
given  by  the  formula 


FIG     192. 


T'  = 


AB 
P' 


(408) 


(6)  Face  of  the  gear. — The  face  of  the  worm  gear  depends  upon 
the  included  face  angle  of  the  worm.  In  Figs.  191  and  193  are 
shown  two  ways  of  making  the  face  of  worm  gears.  The  design 


ART.  271] 


WORM  AND  GEAR  CONSTRUCTION 


377 


shown  in  Fig.  191  is  used  considerably  for  all  ordinary  worm  gears. 
The  face  angle  25  is  chosen  arbitrarily,  and  60  degrees  seems  to 
answer  very  well  for  all  common  proportions,  although  occasion- 
ally 75  degrees  may  be  preferred. 

The  large  diameter  D2  of  the  gear  blank  is  given  by  the  follow- 
ing expression,  provided  the  corners  of  the  teeth  are  left  sharp: 

D2  =  Dl  +  (d  -  2  a)  (1  -  cos  8)  (409) 

in  which  DI  denotes  the  so-called  throat  diameter  and  is  equal 
to  the  pitch  diameter  D  plus  twice  the  addendum  of  the  worm 
teeth. 

The  design  illustrated  by  Fig.  193  is  intended  chiefly  for  worm- 
gears  having  a  large  angle  of  lead.  According  to  the  practice 
of  one  manufacturer  of 
such  gears,  the  magnitude 
of  the  face  angle  26  may 
be  obtained  from  the  for- 
mula 


cos  6 


d  -  3  a 


(410) 


in  which  d  denotes  the 
pitch  diameter  of  the 
worm,  as  shown  in  the 
figure.  The  outside  diam- 
eter D2  of  the  gear  blank  FlG  193 
represented  in  Fig.  193  is 

made  equal  to  the  pitch  diameter  plus  three  times  the  adden- 
dum. The  throat  diameter  DI  is  made  equal  to  the  pitch  diam- 
eter plus  twice  the  addendum. 

271.  Sellers  Worm  and  Rack. — On  planers  and  large  milling 
machines,  the  table  is  driven  by  a  worm  and  rack.     The  teeth 
of  the  rack  are  cut  straight  across  and  not  at  an  angle;  hence  the 
axis  of  the  worm  must  be  set  over  through  an  angle  equal  to  the 
helix  angle.     The  worm  runs  in  an  oil  bath  and  proper  thrust 
bearings  are  provided  to  take  care  of  the  thrust  in  either  direction. 
This  form  of  worm  and  rack  drive  was  introduced  by  the  Wm. 
Sellers  Co.  on  its  planers  and  later  on  it  was  adopted  by  several 
manufacturers  of  large  milling  machines. 

272.  Worm-gear  Mounting. — Generally  speaking,  all  worm- 
gear  transmissions  should   be  mounted  in  a  dustproof  casing 


378 


WORM-GEAR  MOUNTING 


[CHAP.  XIV 


which  permits  either  the  worm  or  the  gear  to  run  in  an  oil  bath. 
In  many  installations  the  worm  is  located  below  the  gear,  while 
in  others  it  must  be  located  above.  In  the  former  case  the  worm 
runs  in  oil,  and  experience  seems  to  indicate  that  such  a  mounting 
gives  the  least  trouble  and  lasts  longer  than  the  second  type. 
There  are,  however,  many  installations  in  which  the  worm  must 
be  mounted  above  the  gear,  and  in  such  cases  the  proper  lubrica- 
tion depends  upon  the  amount  of  oil  carried  to  the  worm  by  the 
gear,  the  lower  segment  of  which  runs  in  the  oil  bath.  Many 
such  drives,  provided  with  the  proper  kind  of  a  lubricant,  are  in 
successful  use. 

From  the  discussion  of  the  various  forces  acting  upon  the 
several  elements  of  a  worm-gear  drive,  it  is  evident  that  the 
thrust  along  both  the  worm  and  worm-gear  shafts  must  be  taken 


FIG.  194. 

care  of  by  suitable  thrust  bearings.  Figs.  189,  190,  and  194  to 
196,  inclusive,  show  several  ways  of  taking  care  of  the  thrusts 
upon  the  shafts  of  a  worm-gear  transmission.  In  a  drive  in 
which  the  efficiency  is  low  or  of  little  consequence,  the  thrust 
along  the  worm  shaft  is  taken  up  by  one  or  more  loose  washers 
made  of  bronze  or  fiber.  If  more  than  one  washer  is  necessary, 
then  alternate  washers  of  steel  and  bronze  give  satisfactory 
service.  The  shaft  bearings  of  a  drive  of  this  kind  are  generally 
made  of  bronze,  but  a  good  grade  of  babbitt  may  also  be  used. 
On  the  worm-gear  shaft  bronze  or  babbitted  bearings  may  be 
employed,  depending  upon  the  magnitudes  of  the  loads  coming 
upon  the  bearings. 

In  a  drive  in  which  the  efficiency  must  be  made  as  high  as 
possible,  ball  or  roller  bearings  must  be  used.  In  Figs.  194  and 
195  are  shown  two  examples  of  a  motor-truck  rear-axle  worm 
mounting  in  which  ball  bearings  are  used.  The  end  thrust  upon 


ART.  272] 


WORM-GEAR  MOUNTING 


379 


the  worm  shaft,  in  the  design  illustrated  by  Fig.  194,  is  taken  by 
the  double-row  ball  bearing,  and,  at  the  same  time,  this  bearing 
takes  its  share  of  the  transverse  loads  upon  the  shaft.  The 
double-row  ball  bearing  is  mounted  rigidly  as  shown,  while  the 
single-row  bearing  has  its  outer  race  floating,  thus  making  pro- 


FIG.   195. 

vision  for  expansion  of  the  worm  shaft.     The  design  just  de- 
scribed was  originated  by  The  New  Departure  Mfg.  Co. 

The  worm-shaft  mounting  illustrated  by  Fig.  195  employs  the 
type  of  radial  ball  bearing  that  is  capable  of  taking  a  thrust, 
the  magnitude  of  which  is  equal  to  or  greater  than  the  radial 


FIG.  196. 

load  coming  upon  them.  Another  feature  worthy  of  attention 
is  the  fact  that  the  worm  shaft  is  always  in  tension,  no  matter  in 
which  direction  the  thrust  of  the  worm  gear  acts. 

In  Fig.  196  is  shown  another  good  example  of  a  rear-axle  worm- 
gear  transmission,  in  which  Timken  conical  roller  bearings  are 


380 


TANDEM  WORM  GEARS 


[CHAP.  XIV 


used  throughout.  An  inspection  of  the  figure  shows  that  the 
worm  shaft  is  always  in  compression,  and  with  the  rigid  mount- 
ing of  the  roller  bearings  on  this  shaft,  no  provision  is  made  for 
taking  care  of  any  expansion  that  may  occur.  A  mounting  simi- 
lar to  that  shown  in  Fig.  195,  but  using  conical  roller  bearings  in 
place  of  the  ball  bearings,  will  prove  satisfactory.  Not  infre- 
quently, the  worm  shaft  is  mounted  upon  ordinary  radial  ball  or 
roller  bearings  and  the  thrust  is  taken  by  a  double-thrust  ball 
bearing.  A  combination  of  radial  and  thrust  bearings  is  efficient, 
but  is  more  or  less  complicated  and  at  the  same  time  is  more 
expensive  than  the  mountings  discussed  above. 


FIG.  197. 

273.  Tandem  Worm.  Gears/ — In  heavy-duty  elevators,  the 
drum  or  traction  sheave  is  driven  by  means  of  double  worm  gear- 
ing, the  arrangement  of  which  is  shown  in  Fig.  197.  Such  a  drive 
consists  of  right-  and  left-hand  worms  cut  integral  with  the  shaft 
and  mounted  below  the  bronze  worm  gears  with  which  they  mesh. 
The  worm  gears  are,  strictly  speaking,  helical  gears  and  since  they 
are  cut  right  and  left  hand  of  the  same  pitch,  they  readily  engage 
with  each  other.  One  of  these  worm  gears  is  connected  to  the 
hoisting  drum  or  sheave.  It  is  evident  that  a  combination  of 
this  description  practically  eliminates  all  end  thrust  on  the  worm 
shaft,  thus  simplifying  the  arrangement  of  the  bearings  on  this 


ART.  274]    EXPERIMENTAL  RESULTS  ON  WORM  GEARING         381 


shaft.  The  part  of  the  shaft  between  the  worms  is  subjected 
either  to  a  tension  or  a  compression,  depending  upon  the  loading 
on  the  hoisting  drum. 

274.  Experimental  Results  on  Worm  Gearing. — A  consider- 
able number  of  tests  of  worm  gearing  have  been  made  by  various 
investigators  in  order  to  determine  the  probable  efficiency  of  such 
gearing,  also  to  determine  the  relation  existing  between  the  coeffi- 
cient of  friction  and  the  sliding  velocity  of  the  teeth  in  contact. 
Evaluating  equation  (402)  for  a  given  coefficient  of  friction  and 
various  angles  of  lead,  it  will  be  found  that  the  efficiency  varies 
but  little  for  angles  between  30  and  60  degrees.  The  results  ob- 
tained from  the  well-known  experiments  on  worm  gearing  made 
by  Wilfred  Lewis  agree  very  closely  with  those  determined  by 
means  of  (402). 

The  value  of  the  coefficient  of  friction  for  any  particular  condi- 
tion of  speed  and  tooth  pressure  is  somewhat  difficult  to  deter- 
mine. The  experimental  results  obtained  by  Lewis,  Stribeck, 
Bach,  Roser,  and  other  investi- 

TABLE    84. — RESULTS    OF   TESTS   ON 

CAST-IRON  WORM  GEARING 

BY  STRIBECK 


Velocity, 
ft.  per  min. 

Pressure, 
pounds 

Coef.  of  fric- 
tion at  60°C. 

98 

1 

0.061 

196 

I  1,100 

0.051 

294 

j 

0.047 

392 

880 

0.040 

586 

550 

0.030 

784 

350 

0.025 

gators  seem  to  lead  to  the 
following  conclusions:  (1)  The 
coefficient  of  friction  appears 
to  have  its  greatest  value  at 
low  speeds,  also  at  high  speeds. 
(2)  The  coefficient  of  friction 
has  its  lowest  values  at  me- 
dium speeds  (200  to  600  feet 
per  minute).  (3)  The  coeffi- 
cient of  friction  varies  but 
little  for  different  tooth  pres- 
sures. In  Table  84  are  given 

some  of  the  results  obtained  by  Stribeck  from  a  series  of  tests 
on  a  cast-iron  worm  and  gear  having  the  following  dimensions : 
The  gear  Was  9^  inches  in  diameter  and  had  30  teeth.  The 
outside  diameter  of  the  single-thread  worm  was  approximately 
3%  inches,  and  the  tangent  of  the  helix  angle  was  given  as  0.1. 
In  the  design  of  high-efficiency  worm  gearing  as  used  in  motor 
cars,  one  authority  recommends  that  /*  may  be  taken  as  low  as 
0.002;  however,  this  value  appears  rather  low  for  general  use  and 
it  is  believed  that  0.01  will  give  safer  results.  For  designing 
single-thread  worms  of  the  irreversible  or  self-locking  type,  the 
coefficient  of  friction  may  be  assumed  as  0.05. 


382  REFERENCES  [CHAP.  XIV 

The  actual  efficiencies  of  well-constructed  and  properly 
mounted  worms  and  gears,  as  used  on  motor  cars,  are  in  general 
high,  running  above  95  per  cent,  in  many  cases. 

References 

American  Machinist  Gear  Book,  by  C.  H.  LOGUE. 

Spiral  and  Worm  Gearing,  by  Machinery. 

Elements  of  Machine  Design,  by  W.  C.  UNWIN. 

Worm  Gearing,  by  H.  K.  THOMAS. 

Herringbone  Gears,  with  special  reference  to  the  Wuest  System,  Trans. 
A.  S.  M.  E.,  vol.  33,  p.  681. 

The  Design  of  Cut  Herringbone  Gears,  Amer.  Mach.,  vol.  43,  pp.  901  and 
941. 

Power  Transmitted  by  Herringbone  Gears,  Mchy.,  vol.  19,  p.  782. 

Theory  of  Enlarged  Herringbone  Pinions,  Mchy.,  vol.  23,  p.  401. 

The  Transmission  of  Power  by  Gearing,  Ind.  Eng'g  and  Eng'g  Digest,  vol. 
14,  p.  114. 

A  New  Gear — The  Circular  Herringbone,  Amer.  Mach.,  vol.  39,  p.  635. 

Making  Worm  Gears  in  Great  Britain,  Amer.  Mach.,  vol.  36,  p.  739. 

Manufacturing  Hindley  Worms,  Amer.  Mach.,  vol.  41,  p.  149. 

Manufacture  of  Worm  Gearing  by  a  New  Process,  Trans.  Soc.  of  Auto. 
Engr.,  January,  1915. 

Gear  for  Panama  Emergency  Gates,  Amer.  Mach.,  vol.  37,  p.  239. 

Allowable  Load  and  Efficiency  of  Worm  Gearing,  Mchy.,  vol.  17,  p.  42. 

Experiments  on  Worm  Gearing,  Trans.  A.  S.  M.  E.,  vol.  7,  p.  284. 

Worm  Gear,  London  Eng'g,  Aug.  20  and  27,  and  Sept.  3,  1915. 

Worm  Gear  and  Worm  Gear  Mounting,  Inst.  of  Auto.  Engr.,  December, 
1916 


CHAPTER  XV 
COUPLINGS 

A  coupling  is  a  form  o£  fastening  used  for  connecting  adjoin- 
ing lengths  of  shafting  so  that  rotation  may  be  transmitted  from 
one  section  to  the  other.  Couplings  may  be  divided  into  the 
following  general  groups :  (a)  permanent  couplings;  (b)  releasing 
couplings. 

PERMANENT  COUPLINGS 

A  permanent  coupling  is  generally  so  constructed  that  it  is 
necessary  to  partially  or  wholly  dismantle  it  in  order  to  separate 
the  connected  shafts.  Hence,  it  is  evident  that  permanent  coup- 
lings are  only  used  for  joining  shafts  that  do  not  require  frequent 
disconnection.  Permanent  couplings  may  be  grouped  into  the 
following  classes: 

(a)  Couplings  connecting  shafts  having  axes  that  are  parallel 
and  coincident. 

(6)  Couplings  connecting  shafts  having  axes  that  are  parallel 
but  not  coincident. 

(c)  Couplings  connecting  shafts  having  axes  that  intersect. 

(d)  Couplings    connecting    shafts    having    inaccurate    align- 
ments. 

COUPLINGS  FOR  CONTINUOUS  SHAFTS 

Some  of  the  requisites  of  a  good  coupling  for  connecting  con- 
tinuous shafts  are  as  follows: 

1.  It  must  keep  the  shafts  in  perfect  alignment. 

2.  It  must  be  easy  to  assemble  or  dissemble. 

3.  It  must  be  capable  of  transmitting  the  full  power  of  the 
shafts. 

4.  The  bolt  heads  and  nuts,  keys  and  other  projecting  parts 
should  be  protected  by  suitable  flanges,  rims,  or  cover  plates. 

275.  Flange  Coupling. — One  of  the  most  common  as  well  as 
most  effective  type  of  permanent  coupling  for  continuous  shafts 
is  the  plain  flange  coupling  shown  in  Fig.  198.  In  order  to  insure 
positive  shaft  alignment,  one  shaft  should  project  through  its 

383 


384 


FLANGE  COUPLING 


[CHAP.  XV 


flange  into  the  bore  of  the  companion  flange.  Another  effective 
way  of  accomplishing  the  same  purpose  is  to  allow  a  part  of  the 
one  flange  to  project  into  a  recess  in  the  other,  as  shown  in  Fig. 
198.  The  coupling  bolts  must  be  fitted  accurately,  generally  a 
driving  fit,  so  that  each  one  will  transmit  its  share  of  the  torsional 
moment  on  the  shaft.  The  size  of  the  bolts  should  be  such  that 
their  combined  shearing  resistance  will  at  least  equal  the  tor- 
sional strength  of  the  shaft.  In  certain  installations  requiring 
accurate  alignment  of  the  shafts,  the  flanges  of  the  coupling  are 
forced  on  the  shaft  and  are  then  faced  off  in  place. 


FIG.  198. 

Analysis  of  a  flange  coupling. — A  flange  coupling  may  fail 
to  transmit  the  full  torsional  moment  of  the  shaft  from  the 
following  causes:  (1)  The  key  may  fail  by  shearing  or  by  crush- 
ing. (2)  The  coupling  bolts  may  fail  by  shearing  or  by  crush- 
ing. (3)  The  flange  may  shear  off  at  the  hub. 

1.  Failure  of  the  key. — To  prevent  the  key  from  shearing,  its 
moment  of  resistance  about  the  axis  of  rotation  must  at  least 
equal  the  torsional  strength  of  the  shaft.  Using  the  notation 
given  in  Art.  93,  the  relation  between  the  shearing  strength  of 
the  key  and  torsional  moment  T  according  to  (104)  may  be 
expressed  as  follows: 

2T 


bl 


-  dS8 


(411) 


ART.  275]  FLANGE  COUPLING  385 

To  prevent  crushing  of  the  key,  the  moment  of  the  crushing 
resistance  of  the  key  about  the  axis  of  rotation  must  exceed 
slightly  the  torsional  moment  T;  whence,  from  (102) 

u^  m        •  (412) 

2.  Failure  of  the  bolts.  —  In  a  flange  coupling  located  at  a  con- 
siderable distance  from  the  bearings  supporting  the  shaft,  the 
bolts  are  generally  subjected  to  bending  stresses  in  addition  to 
crushing  and  shearing  stresses.  It  is  evident,  therefore,  that 
couplings  should  be  located  near  the  bearings.  In  the  following 
analysis  it  will  be  assumed  that  the  coupling  bolts  are  not  sub- 
jected to  a  cross-bending,  but  only  to  shearing  and  crushing 
stresses.  Equating  the  shearing  resistance  of  the  bolts  to  the 
load  coming  upon  them,  we  obtain  the  relation 

a  >  2  Ap^>  (413) 

\Trne£>8 

in  which  a  denotes  the  diameter  of  the  bolts,  n  the  number  of 
bolts  used  in  the  coupling,  and  e  the  diameter  of  the  bolt  circle. 
Instead  of  failing  by  a  shearing  action,  the  bolts  as  well  as  the 
flange  may  fail  by  crushing;  whence  we  obtain  the  relation 

(414) 


in  which  /  denotes  the  thickness  of  that  part  of  the  flange  through 
which  the  bolts  pass. 

3.  Shearing  off  of  the  flange.  —  The  coupling  may  fail  due  to  the 
shearing  of  the  flange  where  the  latter  joins  the  hub.     To  prevent 
this  failure  the  moment  of  the  shearing  resistance  of  the  flange 
must  at  least  equal  the  torsional  moment  transmitted  by  the 
shaft.     Hence,  it  follows  that 

«  *  Hr  (415) 

in  which  c  denotes  the  diameter  of  the  hub,  and  S's  the  allowable 
shearing  stress  in  the  material  of  the  coupling. 

4.  Proportions  of  flange  couplings.  —  In  order  that  a  flange  coup- 
ling may  transmit  the  full  torsional  strength  of  the  shaft  to  which 
it  is  connected,  the  various  relations  derived  above  must  be  satis- 
fied.    The  analysis  of  the  stresses  just  referred  to  is  only  made  in 
special  or  unusual  cases.     For  the  common  flange  coupling  used  on 


386 


COMPRESSION  COUPLING 


[CHAP.  XV 


line-  and  counter-shafts,  it  is  unnecessary  to  make  an  investiga- 
tion of  the  stresses  in  the  various  parts,  as  the  proportions  of  such 
couplings  have  been  fairly  well  established  by  several  manu- 
facturers. However,  no  uniform  proportions  of  flange  couplings 
have  as  yet  been  proposed  for  adoption  as  a  standard.  In  Table 
85  are  given  the  proportions  of  a  series  of  flange  couplings  recom- 
mended by  the  Westinghouse  Electric  and  Mfg.  Co.,  and  these 
represent  good  average  practice.  The  dimensions  listed  in  Table 
85  refer  to  the  flange  coupling  shown  in  Fig.  198. 

276.  Marine  Type  of  Flange  Coupling. — The  type  of  flange 
coupling  shown  in  Fig.  199  is  used  chiefly  in  marine  work  where 

great  strength  and  reliability  are 
of  the  utmost  importance.  The 
fitting  of  this  form  of  coupling  is 
done  with  considerable  care;  for 
example,  the  bolt  holes  are 
always  reamed  after  the  flanges 
are  placed  together,  thus  insur- 
ing perfectly  fitted  bolts,  each  of 
which  will  transmit  its  full  share 
of  the  torsional  moment  upon 
the  shaft.  The  method  of 
analyzing  the  stresses  and  arriv- 
ing at  the  dimensions  of  the  vari- 
ous parts  of  a  marine  flange  coup- 
ling is  similar  to  that  given  for  the  common  flange  coupling. 

277.  Compression  Coupling. — (a)   Clamp  coupling. — A  form 
of  coupling  used  extensively  at  present  on  shafts  of  moderate 
diameter,  say  up  to  approximately  5  inches,  is  shown  in  Fig.  200. 
It  is  commonly  called  a  compression  or  clamp  coupling.     The 
two  halves  of  the  clamp  coupling  are  planed  off,  and  after  the 
bolt  holes  are  drilled,  the  halves  are  bolted  together  with  strips 
of  paper  between  them  and  bored  out  to  the  desired  size.     After 
the  boring  operation,  the  strips  of  paper  are  removed.     When 
the  coupling  is  fastened  to  the  shaft,  the  small  opening  between 
the  two  halves,  due  to  the  removal  of  the  paper,  permits  the  draw- 
ing up  of  the  bolts,  and  a  clamping  action  on  the  shaft  is  thus  pro- 
duced.    The  square  key  used  in  connection  with  a  clamp  coupling 
is  generally  made  straight  and  is  fitted  only  at  the  sides.     This 
coupling  may  be  put  on  and  removed  very  easily,  and  it  has  no 


FIG.  199. 


ART.  275]          WESTINGHOUSE  FLANGE  COUPLINGS 


387 


CO   CO   CO   GO   00   00 


rH\    MX  r-K      t~\   r-K  MX    *H\  ,H\ 

rH    j-H    <M      (N      (N     CO      TtH^lOCOl^OO 


(N      CO  \CO  CO 

\CO    XrH    rH\    \00    \rH    \<M 
05\    lrt\    rH       COX    t>\   r-iX 


CD      CD      C<)        (N  C<)        CD      CD  COCO 

XrH    \rH    \M      \CO       \00    \CO       \rH    \-H    \<#    \rH    XrH    \00 

rH\  rH\   M\     M\     rH\  lrt\     M\  M\  i-K  IO\  1OX  M\ 


\rH    \N    \X    rH\ 
t-\   rH\    1O\   rH 


XT*  \r}(  \M     XrH     MX  iO\     t~\  XOO  Xj*  XpO  XrH 

r-K    r-K    05\      ItK      rH        rH          rH        USX   MK    t~K   r-K 


vrH  CO        CD 

«s\    \rH  XrH  xc<i  x^H  XrJi  XT*  \N 

H         rH\   IOK  rH\   MX    (M  rH\   MX   r-K 

rH     TH  rH     rH  <N     <M     CO 


CD      \rH 

XpO    XPO    \rH     rH\      \« 
rH\   MK    0>\      rH         rHX 


rH        C0\    «5\   COX 

(M     CO     CO 


io\  ioK    M\ 
(M    (M     CO 


oo  o 


CN  (M      CD 

\M  \M    XrH 

CO\  \-ijl  \00     O5\  MX  v 

rH       MK  MX      rH       rH  r 

<M<N  CO     ^*IO     t^OOOiOCO 


XT*  \IM  \oo    \N    vj(  XT*  \e<i  \T*         XCN 

i-f\  i-i\  t~\    i-K    M\  i-K  i-K  MK         i-K 

<NiM<N     CO     T^CO     J>OOC3'-(COCO 


(N    CO     CO 


GO   O   <M    CO   CO   O 


00   O 


lOCOt^      00      rHCO      COOSi-H^OOCO 
J-H.-I       rHrHINlMlNCO 


CO   t>-   00   O 


388 


CLAMP  COUPLINGS 


[CHAP.  XV 


projecting  parts  that  are  liable  to  injure  workmen.  In  Table 
86  are  given  the  general  dimensions  of  a  series  of  sizes  of  the  clamp 
coupling  illustrated  in  Fig.  200. 


Q..G3 


I         I 


FIG.  200. 
TABLE  86. — DIMENSIONS  OF  CLAMP  SHAFT  COUPLINGS 


Shaft 
diameter 


Dimensions 


Diam. 
of  bolts 


Key 


We 


2%  6 

2K6 
2% 

2We 

3%  6 

3K6 
3% 


4% 


6 

7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
18 
20 


4K 


Ke 


8 

8% 
9 
9% 

10K 


3% 
4  ' 

4K 
4K 
4% 
5 


We 


We 


We 


(6)  Nicholson  compression  coupling. — Another  form  of  the  so- 
called  compression  coupling  is  shown  in  Fig.  201.  This  coupling 
requires  no  cutting  of  key  ways  in  the  shafts  that  are  to  be  con- 
nected together.  It  consists  of  two  flanged  hubs  having  tapered 
bores  which  do  not  run  clear  through  the  hub,  but  terminate  a 
short  distance  from  the  outer  end  as  shown  in  the  figure.  Double- 
tapering  steel  jaws  are  fitted  into  the  tapered  bore  and  held  in 


ART.  278] 


ROLLER  COUPLING 


389 


proper  position  by  the  key-seats  or  slots  cut  into  the  end  of  the 
hub.  These  jaws  are  machined  on  the  inner  faces  to  a  radius 
a  trifle  less  than  the  radius  of  the  shaft,  thus  forming  a  positive 
grip  on  the  shaft  when  the  two  flanges  are  drawn  together 
by  the  bolts.  The  adjustment  of  the  coupling  is  always  con- 
centric and  parallel.  No  keys  are  required,  thus  saving  the  cost 


FIG.  201. 

of  cutting  the  key-seat  in  the  shaft  and  of  fitting  the  key.  The 
coupling  illustrated  in  Fig.  201  is  manufactured  by  W.  H. 
Nicholson  and  Co.  of  Wilkes-Barre,  Pa. 

278.  Roller  Coupling. — In  Fig.  202  is  shown  a  form  of  shaft 
coupling  in  which  steel  rollers  are  used  for  gripping  the  shaft.     As 


FIG.  202. 

shown  in  the  figure,  the  coupling  consists  of  a  cylindrical  sleeve 
with  two  eccentric  chambers  on  the  inside.  Each  of  these 
chambers  contains  two  steel  rollers,  held  parallel  to  each  other  by 
a  light  wire  frame.  With  the  rollers  located  in  the  largest  part  of 
the  eccentric  chambers,  the  coupling  may  easily  be  slipped  over 
the  end  of  the  shaft.  A  slight  turn  of  the  coupling  in  either  direc- 


390 


OLDHAM'S  COUPLING 


[CHAP.  XV 


tion  forces  the  rollers  up  the  inclined  sides  of  the  eccentric 
chamber  thereby  locking  the  coupling  to  the  shaft.  Since  no 
screws,  bolts,  pins,  or  keys  are  used  with  this  coupling,  no  tools 
are  needed  in  applying  it  to  a  shaft.  Due  to  the  smooth  exterior, 
the  roller  coupling  shown  in  Fig.  202  insures  freedom  from  acci- 
dent to  workmen. 

COUPLINGS  FOR  PARALLEL  SHAFTS 

279.  Oldham's  Coupling. — When  two  shafts  that  are  parallel, 
but  whose  axes  are  not  coincident,  are  to  be  used  for  transmitting 
power,  a  form  of  connection  known  as  Oldham's  coupling  is  used. 


FIG.  203. 

The  constructive  features  of  such  a  coupling  are  shown  in  Fig. 
203.  It  consists  of  two  flanged  hubs  c  and  d  fastened  rigidly  to 
the  shafts  a  and  b.  Between  these  flanges  is  a  disc  e,  which 
engages  each  flanged  hub  by  means  of  a  tongue  and  groove  joint, 
thus  forming  a  sliding  pair  between  them.  With  this  form  of 
coupling,  the  angular  velocity  of  the  shafts  a  and  6  remains  the 
same. 

Parallel  shafts  may  also  be  connected  by  two  universal  joints 
in  place  of  an  Oldham's  coupling. 


COUPLINGS  FOR  INTERSECTING  SHAFTS 

280.  Universal  Joint. — For  shafts  whose  axes  intersect,  a  form 
of  connection  known  as  Hooke's  coupling  is  frequently  used.  A 
more  familiar  name  for  this  coupling  is  universal  joint.  In  Figs. 
204  to  207,  inclusive,  are  shown  four  types  of  universal  joints. 
The  type  of  joint  illustrated  by  Fig.  204  consists  of  two  U-shaped 
yokes  which  are  fastened  to  the  ends  of  the  shafts  that  are  to  be 
connected  together.  Between  these  yokes  is  located  a  cross- 


ART.  280] 


UNIVERSAL  JOINT 


391 


shaped  piece,  carrying  four  trunnions  which  are  fitted  into  the 
bearings  on  the  U-shaped  yokes.  The  joint  shown  in  Fig.  204 
is  manufactured  by  the  Baush  Machine  Tool  Co.,  and  is  well 


FIG.  204. 
TABLE   87. — PROPORTIONS   OF   BOCORSELSKI'S  UNIVERSAL  JOINT 


Size 


Dimensions 


Diameters 


10 


IK 

2 

2i  KG 


K 


TIG 

KG 


15/B2 


KG 
KG 


'/8 
1 

1K6 


We 


7 

9 

10% 


iK6 
1K6 


2^ 


0.076 

0.1065 

0.167 

K 

KG 


716 

% 


lHe 


0 . 0465 
0.0595 
0.096 


KG 


H 
H 


K 


adapted  for  machine-tool  service  as  found  on  multiple  drills. 
In  Table  87  are  given  general  dimensions  of  the  Bocorselski's 
patent  universal  joint  shown  in  Fig.  204. 


392 


UNIVERSAL  JOINT 


[CHAP.  XV 


The  coupling  shown  in  Fig.  205  is  intended  for  heavy  service, 
as  the  two  yokes  and  the  center  cross  are  made  of  hard  bronze, 
while  the  screws  are  made  of  nickel  steel.  The  maximum 
angular  displacement  of  this  joint  is  limited  to  25  degrees. 


FIG.  205. 


The  universal  joint  in  one  form  or  other  is  used  extensively  in 
motor-car  construction.  In  Fig.  206  is  shown  a  joint  designed  by 
the  Merchant  and  Evans  Co.  of  Philadelphia,  Pa.  The  coupling 


FIG.  206. 
TABLE  88. — DIMENSIONS  OF  MERCHANT  AND  EVANS  UNIVERSAL  JOINTS 


Horse 
power 
rating 

Size  of 
shaft 

Dimensions 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

35 

1M 

5H 

±H 

3 

2H 

2K 

IK 

2H 

2 

m 

K 

K 

35-80 

IK 

1M 

6 

3>£ 

3H 

sy* 

2H 

2y2 

2H 

IK 

i 

X 

ART.  280] 


UNIVERSAL  JOINT 


393 


consists  of  a  flanged  hub  to  which  is  attached  a  ring  having  radial 
slots.  The  flanged  hub  is  made  of  machine  steel  and  the  slotted 
ring  of  a  high-carbon  steel.  Into  the  radial  slots  of  the  ring  are 
fitted  the  projecting  arms  or  teeth  of  the  spider  which  is  also 
made  from  a  high-carbon  steel.  On  the  enlargement  of  the  hub 
of  the  spider  is  formed  a  spherical  surface  which  fits  accurately 
into  a  housing,  the  latter  being  fastened  by  bolts  to  the  slotted 
ring  and  the  flanged  hub.  Spherical  centering  caps  are  fitted  to 
the  inside  faces  of  the  flanged  hub  and  spider.  All  of  the  spherical 
surfaces  have  the  same  center,  which,  for  the  design  shown,  is 
located  on  the  common  center  line  of  the  two  shafts.  The  maxi- 
mum movement  out  of  true  alignment  that  is  permissible  with 
the  style  of  coupling  shown  in  Fig.  206  is  plus  or  minus  4  degrees. 
Table  88  gives  general  dimensions  of  two  sizes  of  this  coupling, 
the  smaller  of  which  is  capable  of  transmitting  35  horse  power 
and  the  larger,  80  horse  power. 


FIG.  207. 


In  Fig.  207  is  shown  another  design  of  universal  coupling  fre- 
quently found  on  motor  cars.  The  constructive  details  are  shown 
more  or  less  clearly  in  the  figure  and  hence  no  further  description 
is  necessary. 


COUPLINGS  FOR  SHAFTS  HAVING  INACCURATE 
ALIGNMENTS 

Frequently  it  is  necessary  to  connect  shafts  in  which  slight 
deviations  in  alignment  must  be  taken  care  of,  as  for  example  in 
connecting  a  prime  mover  to  a  generator,  or  an  electric  motor 
to  a  centrifugal  pump,  blower,  or  generator.  For  a  satisfactory 
connection,  flexible  couplings  are  used.  Several  forms  of  flexible 
couplings  are  now  used  by  various  manufacturers,  and  the 


394 


LEATHER-LINK  COUPLING 


[CHAP.  XV 


following  are  selected  as  typical  illustrations  of  the  different 
types. 

281.  Leather-link  Coupling. — In  Fig.  208  is  shown  a  leather- 
link  flexible  coupling  manufactured  by  The  Bruce  Macbeth 
Engine  Co.  of  Cleveland,  Ohio.  It  consists  of  two  flanged  hubs 
connected  together  by  leather  links  as  shown  in  the  figure.  The 
links  are  held  securely  by  bolts,  which  in  turn  are  fastened  to  the 
flanges  so  that  one  end  of  the  links  is  anchored  to  the  one  flange 
while  the  other  end  is  anchored  to  the  other  flange.  The  torque 
of  one  shaft  is  transmitted  to  the  other  through  the  combination 
of  flanges,  links,  and  bolts.  In  order  to  obtain  the  desired  flexi- 
bility, alternate  holes  in  the  flanges  are  made  larger  so  as  to  per- 
mit sufficient  play  for  the  enlarged  washers  used  on  the  bolts. 


FIG.  208. 


TABLE  89. — DATA  PERTAINING  TO  LEATHER  LINK  COUPLINGS 


Dimensions 

Bore 

Max.  h.p.  at 

Maximum 

Weight, 

d 

100  r.p.m. 

r.p.m. 

Ib 

1 

2 

3 

H 

1.5 

2,400 

15 

5 

5% 

2 

l«e 

2 

2,000 

25 

6 

7 

2^ 

1% 

6 

1,800 

65 

8 

10K 

4 

1% 

10 

1,600 

110 

10 

16 

6 

2^6 

15 

1,500 

210 

13 

20 

8 

2% 

30 

1,250 

335 

15 

24 

10 

3He 

50 

1,000 

560 

18 

29 

12 

4Ke 

100 

850 

1,270 

26 

34 

14 

51A 

200 

750 

1,790 

30 

40 

16 

The  leather  used  for  the  links  is  made  from  selected  hides  and  is 
treated  by  a  special  tanning  process  so  as  to  increase  the  strength 


ART.  282] 


LEATHER-LACED  COUPLING 


395 


and  flexibility.  According  to  one  prominent  manufacturer  of 
leather-link  couplings  the  working  stress  for  the  links  may  be 
taken  as  400  pounds  per  square  inch.  Due  to  the  low  first  cost 
of  leather-link  couplings,  the  General  Electric  Co.  recommends 
their  use  on  all  shafts  up  to  and  including  2  inches  in  diameter. 
For  shafts  from  2  to  3}-^  inches  in  diameter,  either  the  link 
type  or  the  leather-laced  type  may  be  used.  In  Table  89  are 
given  general  dimensions  and  other  data  pertaining  to  the 
coupling  shown  in  Fig.  208. 

282.  Leather-laced  Coupling. — The  leather-laced  flexible  coup- 
ling shown  in  Fig.  209  consists  of  two  cast-iron  flanges  upon 
which  are  bolted  steel  rings.  An  endless  leather  belt  is  laced 


FIG.  209. 

through  a  series  of  slots  that  are  formed  in  the  rim  of  these 
steel  rings.  The  construction  used  offers  a  ready  means  of 
disconnecting  the  machines  without  unlacing  the  belt.  As 
may  be  seen  in  Fig.  209,  disconnection  is  ^accomplished  by  simply 
removing  the  cap  screws  that  fasten  the  outer  steel  ring  to  the 
central  flange.  According  to  the  General  Electric  Co.,  the  manu- 
facturers, this  coupling  is  recommended  when  the  shafts  to  be 
connected  are  more  than  3J^  inches  in  diameter.  The  belting 
used  is  made  from  a  specially  prepared  leather  capable  of 
carrying  a  working  stress  of  400  pounds  per  square  inch  of 


396 


FRANCKE  COUPLING 


[CHAP.  XV 


section.     In  Table  90  are  given  general  dimensions  and  other 
data  pertaining  to  the  laced-belt  coupling  shown  in  Fig.  209. 

TABLE  90. — DATA  PERTAINING  TO  LEATHER  LACED  COUPLINGS 


Bore 
d 

Max.  h.p. 
at 
100  r.p.m. 

Max. 
r.p.m. 

Weight, 
lb. 

Dimensions 

Key 

1 

2 

3 

4 

5 

6 

2Si 

16 

1,200 

160 

15H 

10 
12 

5 
6 

4^6 

41Me 

4?i 

MXH 

3 
3H 

27.7 

900 

263 
256 

18>£ 

5^6 

5*^6 

5Me 

^X^ 

4 
4H 

66 

750 

494 

482 

24H 

14 

8 

6% 

6^6 

634 

1X1 

5 
5^ 

128 

600 

883 
868 

30H 

10 

10 

12 

7^6 

71K6 

7>£ 

1MX1K 

SH 

450 

1,329 
1,307 

37 

IS 

8% 

8i  He 

7H' 

iHxiM 

?M       !      352 

350 

2,076 
2,046 

43 

20 

14 

9^6 

9^6 

8^ 

8H 

526 

300 

2,767 
2,727 

49 

24 

28 

16 

11^6 

H1M6 

9K 

iy*xiH 

9 
9K 

748 

250 

3.917 
3,865 

55 

18 

13^6 

13% 

9iKe 

1%X2 

10 

1,027 

200 

5,120 

61 

32 

20 

15^{6 

15% 

lOMe 

In  general,  flexible  couplings  using  leather  as  the  connecting 
medium  are  not  recommended  for  places  where  dampness  or 
oil  would  affect  the  leather.  Neither  should  they  'be  used  when 
flying  dust  or  grit  are  liable  to  injure  the  leather  links  or  lacing. 
It  is  generally  assumed  that  the  leather  connectors  afford  suffi- 
cient insulation  between  the  two  halves  of  the  coupling  when  the 
latter  is  used  in  connection  with  electric  motors  or  generators. 

283.  Francke  Coupling. — The  type  of  flexible  couplings  dis- 
cussed in  the  two  preceding  articles  transmit  power  from  the 
driving  to  the  driven  member  by  means  of  a  fibrous  material. 
Couplings  having  soft  rubber  buffers  between  interlocking  arms 
of  two  cast-iron  spiders  have  also  been  used  successfully.  Re- 
cently a  form  of  coupling  known  as  the  Francke  flexible  coupling 
in  which  a  pair  of  flanges  are  connected  by  flexible  steel  pins 
was  placed  on  the  market.  The  constructive  details  of  this 
coupling  are  shown  clearly  in  Fig.  210.  The  so-called  pins  are 
built  up  of  a  series  of  tempered-steel  plates  having  a  slotted  hole 


ART.  283] 


FRANCKE  COUPLING 


397 


at  each  end  through  which  a  hardened-steel  pin  passes.  By 
means  of  these  pins,  the  ends  of  the  tempered  plates  are  held  in 
steel  yokes  which  are  fastened  to  the  rims  of  the  flanges  by  means 
of  cap  screws,  as  shown  in  Fig.  210.  In  the  smaller  sizes  of  the 
Francke  coupling,  the  ends  of  the  steel  yokes  and  the  inner 
surfaces  of  the  coupling  flanges  have  grooves  into  which  steel 
rings  are  sprung,  thus  holding  the  tempered  plates  in  a  radial 
position. 

Any  flange  coupling  connecting  two  shafts  that  are  out  of 
alignment  will  run  open  on  the  one  side  and  closed  on  the  other. 


FIG.  210. 

The  endwise  motion  due  to  this  opening  and  closing  action  of  the 
flanges  is  provided  for,  in  the  Francke  coupling,  by  the  slotted 
holes  near  the  ends  of  the  tempered-steel  plates. 

In  Table  91  are  given  general  dimensions,  net  weights,  permis- 
sible speeds,  and  approximate  horse  powers  pertaining  to  the 
commercial  sizes  of  the  coupling  shown  in  Fig.  210.  The  follow- 
ing directions  for  selecting  the  proper  size  of  coupling  for  any 
desired  service  are  recommended  by  the  manufacturers  of  the 
Francke  coupling. 


398 


FRANCKE  COUPLING 


[CHAP.  XV 


(a)  From  Table  91,  select  the  smallest  coupling  having  a  maxi- 
mum bore  large  enough  to  receive  the  largest  shaft  to  be 
connected. 

(6)  For  the  installation  under  consideration,  determine  the 
horse  power  transmitted  per  100  revolutions  per  minute. 

TABLE    91. — DATA    PERTAINING    TO    THE    FRANCKE    COUPLING — HEAVY 

PATTERN 


Size 
No. 

Max. 
bore 

Max.h.p. 
at  100 
r.p.m. 

Max.  r.p.m. 

Weight 

Dimensions 

Key 

Cast  iron 

Steel 

1 

2 

3 

4 

5 

3% 

% 

1.33 

4,000 

10,000 

8.5 

3% 

4% 

UK. 

2^6 

„„ 

K.XK. 

4 

1% 

2 

11 

4 

5% 

2<Ke 

2^6 

KXK 

4% 

\H 

2.75 

14 

4% 

5% 

2% 

2% 

,». 

5 

6 

7 

2 

2% 

3.75 
6.5 
9 

3,500 
3,100 
2,500 

8,500 
7,600 
6,400 

20 
35 

45 

5 
6 

7 

6% 

3 

3% 
4% 

2^6 

KeXMe 

HXH 

8% 

3 

28 

2,150 

5,400 

70 

8% 

7% 

5% 

3^6 

1% 

%X%' 

10 

3% 

65 

1,800 

4,600 

115 

10 

6% 

3% 

HXH 

12 

4% 

91 

1,500 

3,800 

210 

12 

9% 

8% 

4Ke 

%  x% 

15 

6 

145 

1,200 

3,000 

385 

15 

11% 

11 

5ffe 

2% 

%x% 

18 

7% 

210 

1,000 

2,500 

555 

18 

13% 

13% 

6% 

1  X% 

22 

10 

300 

800 

2,000 

1,000 

22 

16% 

17% 

8%  6 

1%-Xl 

24 

27 

9 
11 

750 
1,000 

750 
700 

1,900 
1,700 

1,250 
1,650 

24 

27 

22% 

16% 
19% 

9 
11 

4% 

33 

14 

2,500 

575 

1,400 

3,330 

33 

26% 

24 

13 

5% 

TABLE  92. — FACTORS  FOR  VARIOUS  CLASSES  OF  SERVICE 


Class  of  service 


Steam  turbines  connected  to  centrifugal  pumps  and  blowers 

Turbines  and  motors  connected  to  generators 

Motors  connected  to  centrifugal  pumps  and  blowers 

Motors  connected  to  wood-working  machinery 

Motors  connected  to  grinders,  conveyors,  screens,  and  beaters  with 

no  pulsations 

Motors  connected  to  crushers,  tubemills,  and  veneer  hogs 

Gas  and  steam  engines  connected  to  machines  carrying  a  uniform 

load . . 


Engines  connected  to  fans 

Motors  connected  to  single-cylinder  compressors 

Rolling  mills 

Motors  connected  to  mine  hoists,  elevators  or  cranes. . 


ART.  284] 


NUTTALL  COUPLING 


399 


(c)  From  Table  92,  select  the  factor  for  the  class  of  service  for 
which  the  coupling  is  intended  and  multiply  it  by  the  horse 
power  transmitted  per  100  revolutions  per  minute. 

(d)  Compare  the  horse  power  determined  in  (c)  with  the  horse 
power  rating  of  the  coupling  selected  in  (a)  above.     In  case  the 
latter  is  less  than  the  former,  select  a  larger  coupling  having  the 
desired  rating. 

(e)  If  the  required  speed  is  in  excess  of  that  listed  for  the  cast- 
iron  coupling,  use  a  steel  coupling. 

284.  Nuttall  Coupling. — The  Nuttall  coupling  illustrated  in 
Fig.  211  differs  considerably  from  those  discussed  in  the  preced- 
ing articles,  in  that  the  power  is  transmitted  through  the  medium 


FIG.  211. 


of  helical  springs  c.  These  springs  with  the  inserted  case-hard 
ened  plugs  d  are  fitted  into  pockets  between  the  twin-arms  of 
the  spider  6.  The  casing  a  is  provided  with  a  series  of  lugs  that 
fit  loosely  in  the  twin-arms  of  the  spider  and  also  bear  against 
the  spring  plugs  d.  It  is  evident  that  with  the  construction 
shown  in  the  figure  this  coupling  can  transmit  power  in  either 
direction,  and,  furthermore,  that  the  springs  are  always  in  com- 
pression. The  clearance  between  the  ends  of  the  spring  plugs  is 
made  slightly  less  than  the  maximum  deflection  of  the  spring; 
therefore,  a  sudden  overload  cannot  break  the  springs.  The 
coupling  has  a  smooth  exterior,  hence  there  is  not  much  danger 
of  injury  to  workmen. 


400 


CLARK  COUPLING 


[CHAP.  XV 


285.  Clark  Coupling. — An  interesting  form  of  flexible  coupling 
that  was  placed  upon  the  market  recently  is  the  Clark  coupling 
shown  in  Fig.  212.  It  consists  of  two  hubs  upon  the  flanges  of 
which  are  cut  a  number  of  special  teeth.  Over  these  teeth  is 
fitted  a  roller  chain  as  shown  in  the  figure.  The  teeth  are  cut 


FIG.   212. 

accurately  so  that  all  of  the  rollers  in  the  chain  are  in  contact 
with  the  teeth,  thus  insuring  an  equal  distribution  of  the  load 
transmitted  by  the  coupling.  Side  clearance  is  provided  between 
the  chain  and  the  teeth,  thus  permitting  the  two  halves  of  the 
coupling  to  take  care  of  any  slight  angular  displacement  of  the 


FIG.  213. 

shafts.  The  chain  is  provided  with  a  master  link  which  may 
be  removed  quickly  in  case  it  is  desired  to  run  each  shaft 
independently. 

286.  Kerr  Coupling. — A  type  of  flexible  coupling  particularly 
well  adapted  to  very  high  rotative  speeds  is  that  shown  in  Fig. 
213.  It  was  developed  by  Mr.  C.  V.  Kerr  for  use  in  connecting 


ART.  286] 


KERR  COUPLING 


401 


steam  turbines  to  centrifugal  pumps  and  blowers.  In  order  to 
make  it  possible  to  use  this  coupling  at  high  speeds,  the  dimen- 
sions are  all  kept  down  to  a  minimum  by  making  the  various 
parts  of  crucible  cast  steel.  The  through  keys  or  cotters  are 
made  of  tool  steel  and  tempered.  Due  to  the  arrangement  of  the 
through  keys  at  right  angles  to  each  other,  the  two  shafts  to  be 
connected  may  be  out  of  alignment  to  a  considerable  extent.  To 
prevent  serious  wear  of  the  various  parts  and  to  eliminate  excess- 
ive noise,  the  coupling  is  filled  with  a  heavy  machine  oil,  or  grease 
and  graphite.  To  design  a  coupling  of  this  kind  the  following 
method  of  procedure  is  suggested: 

(a)  Design  the  shaft  so  that  it  will  readily  transmit  the  re- 
quired horse  power  at  the  specified  speed. 


FIG.    214. 


(6)  Design  the  cross-key  so  that  it  will  be  amply  strong  against 
failure  due  to  crushing,  shearing,  and  bending. 

(c)  Design  the  shell  so  that  it  will  transmit  the  torsional  mo- 
ment of  the  shaft.  The  key-ways  in  the  shell  should  be  investi- 
gated for  crushing. 

287.  Rolling-mill  Coupling. — Frequently,  flexible  couplings  are 
required  in  places  where  considerable  grit,  water,  steam,  etc.,  are 
present,  and  where  noise  is  not  objectionable;  for  example,  in 
a  rolling  mill.  For  such  and  other  heavy  service,  the  rolling- 
mill  type  of  flexible  coupling  shown  in  Fig.  214  is  recommended. 
When  the  load  transmitted  is  practically  constant,  a  rolling-mill 
coupling  will  not  be  excessively  noisy  and  good  results  may  be 
expected. 


402 


JAW  CLUTCH 


RELEASING  COUPLINGS 


[CHAP.  XV 


A  releasing  coupling,  or  clutch,  as  it  is  commonly  called,  is 
so  constructed  that  the  connected  shafts  may  be  disengaged  at 
will.  From  this  statement  it  should  not  be  inferred  that  clutches 
are  used  for  connecting  shafts  exclusively,  as  they  are  also  used 
for  engaging  pulleys,  gears  and  other  rotating  parts.  Clutches 
may  be  divided  into  two  classes  namely:  (a)  Positive  clutches; 
(b)  friction  clutches.  The  latter  class  will  not  be  discussed  in  this 
chapter,  but  will  be  taken  up  in  detail  in  the  following  chapter. 

288.  Positive  Clutch. — The  simplest  form  of  positive  clutch 
is  the  jaw  clutch  shown  in  Fig.  215(a).  One  part  of  the  clutch 
is  keyed  or  pinned  rigidly  to  the  shaft  while  the  other  part  is 
splined,  thus  permitting  it  to  be  engaged  with,  or  disengaged 
from,  the  first  part  by  sliding  it  along  the  shaft.  The  interlock- 
ing jaws  upon  the  abutting  faces  of  the  clutch  may  have  various 


forms,  as  shown  in  Fig.  215.  The  jaws  of  the  type  shown  in 
(b)  engage  and  disengage  more  freely  than  square  jaws.  The 
jaws  illustrated  by  (c),  (d),  and  (e)  are  intended  for  installations 
where  it  is  necessary  to  transmit  power  in  only  one  direction. 
In  punching  and  shearing  machines  the  types  of  jaws  shown 
by  (a)  and  (e)  are  used  considerably. 

289.  Analysis  of  Jaw  Clutches. — Having  decided  upon  the 
type  of  clutch  to  be  used  for  a  particular  installation,  the  next 
step  calls  for  the  determination  of  the  dimensions  of  the  several 
parts.  In  general,  jaw  clutches  are  designed  by  empirical  rules, 
and  consequently  the  resultant  proportions  are  liberal.  How- 
ever, if  it  is  desired  to  arrive  at  the  proportions  of  a  jaw  clutch 
capable  of  transmitting  a  certain  amount  of  power,  the  following 
analysis  is  suggested: 

(a)  Bore  of  the  sleeves. — The  bore  of  the  sleeves  is  fixed  by  the 
size  of  the  shaft  required  to  transmit  the  required  power. 


ART.  289]  JAW  CLUTCH  ANALYSIS  403 

(6)  Length  of  the  sleeves.  —  If  keys  are  used  for  fastening  the 
sleeves  to  the  shafts,  the  lengths  of  the  sleeves  are  fixed,  in  a 
general  way,  by  the  length  of  keys  required  to  transmit  the  de- 
sired power.  In  connection  with  punching  and  shearing 
machinery,  where  the  clutch  sleeve  is  occasionally  fitted  onto  a 
squared  shaft,  the  length  of  the  sleeve  may  be  assumed  approxi- 
mately equal  to  the  diameter  of  the  shaft. 

(c)  Outside  diameter  of  sleeves.  —  The  outside  diameter  of  the 
sleeves  must  be  such  that  the  safe  shearing  strength  of  the  jaws 
will  exceed  the  pressure  coming  upon  them.  The  pressure  upon 
the  jaws  should  be  calculated  on  the  assumption  that  it  is  con- 
centrated at  the  mean  radius  of  the  jaws. 
Let  A  —  area  of  the  jaw  at  the  root. 

D  =  outside  diameter  of  the  clutch  sleeve. 
S,  =  permissible  shearing  stress  of  the  material. 
T  —  torsional  moment  to  be  transmitted  by  the  clutch. 
d  =  bore  of  the  clutch  sleeve. 
n  =  number  of  jaws  on  the  clutch  sleeve. 

Equating  the  torsional  moment  T,  to  the  moment  of  the 
shearing  resistance  of  the  jaws,  and  solving  for  the  total  re- 
quired shearing  area,  we  obtain  the  following  expression: 

nA  =  /nvo-  (416) 

(D 


Without  introducing  any  appreciable  error,  the  area  nA  may  be 
taken  as  equivalent  to  one-half  the  area  between  the  circles  having 
diameters  equal  to  the  outer  and  inner  diameters  of  the  clutch 
sleeve.  Substituting  for  nA  an  expression  for  the  equivalent 
area  in  terms  of  D  and  d}  we  arrive  at  the  following  relation: 

qo  rr 

(D2  -  d*)(D  +  d)  =  ^-  (417) 

In  determining  the  outer  diameter  D  by  the  use  of  (417),  con- 
siderable time  may  be  saved  by  solving  this  equation  by  trial. 

(d)  Number  and  height  of  jaws.  —  The  number  of  jaws  on 
clutches  depends  upon  the  promptness  with  which  a  clutch  must 
act.  In  punching  and  shearing  machinery,  the  number  of  jaws 
varies  from  two  to  four,  while  in  other  classes  of  machinery  the 
number  of  jaws  may  run  as  high  as  twenty-four. 

The  height  of  the  jaws  must  be  such  that  the  pressure  coming 
upon  them  does  not  exceed  the  safe  crushing  strength  of  the 


404  JAW  CLUTCH  ANALYSIS  [CHAP.  XV 

material  used  in  the  clutch.  The  distribution  of  the  pressure 
upon  the  face  of  the  jaws  depends  upon  the  grade  of  workman- 
ship put  upon  the  clutch  parts.  On  clutches  found  on  the 
modern  machine  tools,  we  may  safely  say  that  the  workmanship 
is  of  such  a  quality  that  the  pressure  upon  the  jaws  may  be 
assumed  as  uniformly  distributed. 

Denoting  the  area  of  the  engaging  face  of  one  jaw  by  the 
symbol  Ac,  and  the  permissible  crushing  stress  of  the  material 
by  Sc,  we  obtain  the  following  relation  by  equating  the  torsional 
moment  T  to  the  moment  of  the  resistance  to  crushing: 


*       n(D  +  d)Sc 

Having  determined  the  area  required  to  prevent  crushing  of  the 
jaw,  the  height  h  of  the  latter  is  given  by  the  following  expression: 

*  =    =-  (419) 


Frequently,  the  height  of  the  jaw  as  determined  by  (419)  is  so 
small  that  it  must  be  increased  in  order  that  the  mating  jaws 
will  hook  together  sufficiently  and  not  be  disengaged  by  any 
jarring  action.  Good  judgment  should  play  an  important  part 
in  arriving  at  the  various  dimensions  of  the  parts  of  a  jaw  clutch. 

References 

Elements  of  Machine  Design,  by  W.  C.  UN  WIN. 

Machine  Design,  Construction,  and  Drawing,  by  H.  J.  SPOONER. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Design  of  Punch  and  Shear  Clutches,  Am.  Mach.,  vol.  36,  p.  991. 

Mechanical  Engineers'  Handbook,  by  L.  S.  MARKS,  EDITOR  IN  CHIEF. 

Bulletin  No.  48  ISA-Couplings,  by  General  Electric  Co. 

The  Universal  Joint,  Am.  Mach.,  vol.  38,  p.  108. 

Friction  Losses  in  the  Universal  Joint,  Trans.  A.  S.  M.  E.,  vol.  36,  p.  461. 

Catalogs  of  Manufacturers. 


CHAPTER  XVI 
FRICTION  CLUTCHES 

290.  Requirements  of  a  Friction  Clutch. — The  object  of 
a  friction  clutch  is  to  connect  a  rotating  member  to  one 
that  is  stationary,  to  bring  it  up  to  speed,  and  to  transmit 
the  required  power  with  a  minimum  amount  of  slippage.  In 
connection  with  machine  tools,  a  friction  clutch  introduces  what 
might  be  termed  a  safety  device  in  that  it  will  slip  when  the  pres- 
sure on  the  cutting  tool  becomes  excessive,  thus  preventing  the 
breakage  of  gears  or  other  parts. 

In  designing  a  friction  clutch,  the  following  points  must  be 
given  careful  consideration: 

(a)  The    materials    forming   the    contact    surfaces    must    be 
selected  with  care. 

(b)  Sufficient  gripping  power  must  be  provided  so  that  the 
load  may  be  transmitted  quickly. 

(c)  In  order  to  keep  the  inertia  as  low  as  possible,  a  clutch 
should  not  be  made  too  heavy.     This  is  very  important  in  high- 
speed service,  such  as  is  found  in  motor  cars. 

(d)  Provision  for  taking  up  wear  should  be  made. 

(e)  Provision  should  be  made  for  carrying  away  the  heat  that 
is  generated  at  the  contact  surfaces. 

(/)  A  clutch  should  be  simple  in  design  and  contain  as  few  parts 
as  possible. 

(gr)  The  construction  should  be  such  as  to  facilitate  repair. 

(h)  The  motion  should  be  transmitted  without  shock. 

(i)  A  clutch  should  disengage  quickly  and  not  "drag." 

(j)  A  clutch  transmitting  power  should  be  so  arranged  that  no 
external  force  is  necessary  to  hold  the  contact  surfaces  together. 

(k)  A  clutch  intended  for  high-speed  service  must  be  balanced 
carefully. 

(I)  A  clutch  should  have  as  few  projecting  parts  as  possible, 
and  such  parts  as  do  project  should  be  covered  or  guarded  so  that 
workmen  cannot  come  into  contact  with  them. 

405 


406  CLUTCH  FRICTION  MATERIALS          [CHAP    XVI 

291.  Materials  for  Contact  Surfaces. — In  order  that  a  material 
may  give  satisfactory  service  as  a  frictional  surface,  it  must  fulfill 
the  following  conditions:  (1)  The  material  must  have  a  high  coef- 
ficient of  friction.  (2)  The  material  must  be  capable  of  resisting 
wear.  (3)  The  material  must  be  capable  of  resisting  high  tem- 
peratures, caused  by  excessive  slippage  due  to  frequent  operation 
of  the  clutch. 

Among  the  materials  met  with  in  modern  clutches  are  the 
following  : 

(a)  Wood. — In  many  clutches  used  on  hoisting  machinery,  as 
well  as  in  some  used  for  general  transmission  purposes  on  line-  and 
counter-shafts,  the  contact  surfaces  are  made  of  wood  and  cast 
iron.  Among  the  kinds  of  wood  that  have  proven  satisfactory 
in  actual  service  are  basswood,  maple,  and  elm. 

(6)  Leather. — The  majority  of  the  cone  clutches  in  use  on 
motor  cars  are  faced  with  leather.  Some  manufacturers  use 
oak-tanned,  while  others  prefer  the  so-called  chrome  leather.  To 
obtain  the  best  service  from  a  leather  facing,  it  should  be  treated 
by  soaking  it  in  castor  oil  or  neat's-foot  oil,  or  boiling  it  in  tallow. 
Before  applying  the  facing  to  the  clutch,  the  treated  leather 
should  be  passed  between  rolls  so  as  to  remove  the  excess  oil  or 
grease.  Leather  facings  should  never  be  allowed  to  become  dry 
or  hard,  or  the  clutch  will  engage  too  quickly.  Leather  that  has 
become  charred  due  to  excessive  slippage  has  very  little  value  as 
a  friction  material. 

(c)  Asbestos  fabric. — At  the  present  time  there  are  upon  the 
market  several  patented  asbestos  fabrics  consisting  mainly  of 
asbestos  fiber.  To  give  it  the  necessary  tenacity  the  asbestos 
fiber  is  woven  onto  brass  or  copper  wires.  Among  the  well- 
known  asbestos  fabrics  used  for  clutches,  as  well  as  for  brakes, 
are  Raybestos,  Thermoid,  and  Non-Burn.  The  first  two  may  be 
obtained  in  thicknesses  varying  from  J£  to  J^  inch,  inclusive,  and 
in  widths  of  1  to  4  inches,  inclusive.  Non-Burn  is  made  in  thick- 
nesses up  to  and  including  1  inch,  and  in  widths  up  to  and  in- 
cluding 24  inches.  Asbestos  fabric  facings  are  used  to  a  limited 
extent  on  cone  clutches,  and  on  a  large  number  of  modern  disc 
clutches.  When  it  is  used  on  the  latter  type  of  clutch,  the  fabric 
may  be  riveted  to  the  driving  or  to  the  driven  discs,  whichever  is 
the  more  economical. 

The  Johns-Manville  Co.  manufactures  an  asbestos-metallic 
block  that  is  giving  excellent  service  on  clutches  and  brakes. 


ART.  291]  CLUTCH  FRICTION  MATERIALS  407 

The  block  is  constructed  of  long-fiber  asbestos,  reinforced  with 
brass  wire  and  moulded  under  an  enormous  pressure  into  any 
desired  shape. 

The  main  advantages  claimed  for  the  use  of  wire-woven  as- 
bestos fabric  and  asbestos-metallic  block  are  the  following : 

1.  Slightly  higher  coefficient  of  friction. 

2.  Ability  to  withstand  high  temperatures. 

3.  May  be  run  dry  or  with  oil. 

4.  Not  affected  by  moisture. 

5.  Ability  to  resist  wear. 

(d)  Paper. — Compressed  strawboard  may  be  used  as  a  fric- 
tion surface  on  clutches  in  which  the  speeds  and  the  pressures 
coming  upon  the  contact  surfaces  are  low.     If  excessive  slippage 
occurs,  the  strawboard  is  liable  to  become  charred  rather  rapidly. 
Vulcanized  fiber,  which  is  nothing  more  than  a  form  of  paper 
treated  chemically,  gives  fairly  good  service  as  a  friction  material 
in  clutches.     It  is  capable  of  withstanding  medium  pressure,  as 
well  as  considerable  slippage. 

(e)  Cork  inserts. — Cork  is  never  used  alone  as  a  friction  ma- 
terial, but  always  in  connection  with  some  other  material  either 
of  a  fibrous  or  a  metallic  nature.     It  is  frequently  used  on  leather- 
faced  cone  and  metallic  disc  clutches,  and  is  generally  in  the  form 
of  round  plugs  or  inserts.     The  surface  covered  by  these  cork 
inserts  varies  from  10  to  40  per  cent,  of  the  total  frictional  area. 
Due  to  the  higher  coefficient  of  friction  of  cork,  a  motor-car 
clutch  equipped  with  cork  inserts  is  capable  of  transmitting  a 
little  more  power  for  the  same  spring  pressure  than  a  similar 
clutch  lined  with  leather;  or  for  the  same  power,  the  spring  pres- 
sure in  the  former  is  less  than  in  the  latter  type  of  clutch.     Cork 
inserts  are  also  used  on  hoisting-drum  cone  clutches  having  wood 
blocks,  and  on  common  transmission  clutches  of  the  disc  type. 
Experience  has  shown  that  they  give  excellent  service.     In  gen- 
eral, the  cork  inserts  are  operative  only  at  low  pressures,  as  in 
engaging  the  clutch.     In  combination  with  the  cork,  the  metal, 
leather,  or  wood  in  which  it  is  imbedded  forms  the  surface  in  con- 
tact after  full  engagement.     Cork  inserts  also  aid  in  keeping  the 
surfaces  lubricated. 

(/)  Metallic  surfaces.' — The  materials  discussed  above  are  all  of 
a  fibrous  nature,  and  are  always  used  in  conjunction  with  a 
metal,  such  as  cast  iron,  steel  casting,  steel,  or  bronze.  Fre- 
quently, cone  clutches  used  on  machine  tools  have  both  cones 


408  CONE  CLUTCHES  [CHAP.  XVI 

made  of  cast  iron,  while  in  other  cases  cast  iron  and  steel  casting 
are  used.  Disc  clutches  using  hard  saw-steel  discs  running  in  oil 
are  advocated  by  some  manufacturers;  others  use  steel  against 
bronze,  cast  iron  against  bronze,  and  cast  iron  against  cast  iron. 
In  all  of  the  clutches  using  the  metal-to-metal  surfaces,  a  liberal 
supply  of  oil  is  furnished  by  some  means  or  other. 

292.  Classification   of  Friction  Clutches. — According  to   the 
direction  in  which  the   pressure  between  the  contact  surfaces 
is  applied,  friction  clutches  may  be  divided  into  two  general 
classes,  as  follows : 

(a)  Axial  clutches,  which  include  all  those  having  the  contact 
pressure  applied  in  a  direction  parallel  to  the  axis  of  rotation. 
This  class  includes  all  types  of  cone  and  disc  clutches. 

(b)  Rim  clutches,  which  include  all  clutches  having  the  con- 
tact pressure  applied  upon  a  rim  or  sheave  in  a  direction  at  right 
angles  to  the  axis  of  rotation. 

AXIAL  CLUTCHES 

A  study  of  the  designs  of  the  clutches  manufactured  by  the 
various  builders  of  transmission  machinery,  machine  tools,  and 
motor  cars,  shows  that  axial  clutches  are  made  in  a  variety  of 
forms.  Such  a  study  leads  to  the  following  classification  of 
axial  clutches:  (1)  cone;  (2)  disc;  (3)  combined  conical  disc. 

CONE  CLUTCHES 

The  cone  clutch  is  without  doubt  the  simplest  form  of  friction 
clutch  that  can  be  devised,  and  if  properly  designed  will  give 
entire  satisfaction.  Two  types  of  cone  clutches  are  commonly 
met  with,  as  follows:  (1)  single-cone;  (2)  double-cone. 

293.  Single -cone    Clutch. — The   elements    of   a  simple  cone 
clutch  are  shown  clearly  in  Fig.  216.     The  clutch  consists  of  a 
cone  b  keyed  rigidly  to  the  shaft  a,  while  a  second  cone  d  is 
fitted  to  the  shaft  c  by  means  of  the  feather  key  e.     This  key 
permits  the  cone  d  to  be  engaged  with  the  cone  b,  thus  trans- 
mitting the  power  from  one  shaft  to  the  other.     The  hub  of  the 
cone  d  is  fitted  with  a  groove  /,  into  which  is  fitted  the  shifter 
collar  operated  by  the  engaging  lever. 

(a)  Machine-tool  cone  clutch. — A  good  example  of  the  use  of  a 
simple  cone  clutch,  applied  to  a  machine  tool,  is  shown  in  Fig. 


ART.  293] 


CONE  CLUTCHES 


409 


FIG.  216. 


FIG.  217. 


410  CONE  CLUTCHES  [CHAP.  XVI 

217.  The  design  shown  is  that  used  on  the  main  driving  pulley 
of  the  Lucas  boring  machine.  The  driving  pulley  a  runs  loose 
on  the  hub  of  the  main  bearing  c,  and  has  bolted  to  it  the  cast- 
iron  cone  6.  The  sliding  cone  d  is  fitted  into  b  and  is  keyed  to 
the  main  driving  shaft  k  by  a  feather  key.  The  cones  are  engaged 
by  means  of  the  sliding  spool  h  and  the  levers  e  and  /.  Several 
helical  springs,  one  of  which  is  shown  in  Fig.  217,  are  placed  into 
holes  drilled  into  the  hub  of  the  cone  d;  the  function  of  these 
springs  is  to  disengage  the  two  cones  when  the  spool  h  is  moved 
toward  the  end  of  the  sleeve  g.  It  is  quite  evident  that  this 
clutch  fulfills  the  important  requirement  met  with  in  machine 
tools,  namely,  compactness  and  simplicity  of  design  and  ease  of 
operation. 

(6)  Motor-car  cone  clutch. — With  the  development  of  the 
modern  automobile,  the  design  of  cone  clutches  was  given  more 
attention,  and  at  the  present  time  approximately  40  per  cent, 
of  the  pleasure-car  manufacturers  are  equipping  their  cars  with 
clutches  of  the  single-cone  type.  In  motor  cars  the  clutch  is 
used  to  connect  the  motor  to  the  transmission,  and  normally 
is  held  in  engagement  by  a  spring  pressure.  This  spring  pressure 
must  be  released  by  the  pedal  when  the  car  is  to  be  stopped  or 
when  speed  changes  are  made  by  shifting  the  gears. 

National  clutch. — In  Fig.  218  is  shown  a  design  of  a  cone 
clutch  used  on  the  National  motor  car.  The  cone  a  with  its 
various  attachments  is  forced  into  the  conical  bore  of  the  fly- 
wheel rim  by  the  pressure  of  the  helical  spring.  To  decrease  the 
weight  of  the  clutch,  the  cone  a  is  made  of  aluminum  having  its 
periphery  faced  with  leather.  The  small  flat  springs  b,  with 
which  the  cone  is  fitted  at  various  points  along  its  periphery, 
provide  the  smooth  and  easy  engagement  so  desirable  in  motor 
cars.  To  prevent  spinning,  the  sliding  sleeve  c  has  fastened  to 
it  a  small  brake  sheave  d  upon  which  a  brake  block  e  acts.  The 
brake  block  is  fitted  to  an  operating  lever  /  which  is  depressed 
when  the  clutch  is  disengaged. 

Cadillac  clutch. — The  clutch  shown  in  Fig.  219  is  that  used 
on  the  old  four-cylinder  Cadillac  motor  car.  It  differs  consider- 
ably from  the  National  clutch  discussed  above.  The  cone  a  is 
made  of  pressed  steel,  and  the  flywheel  instead  of  having  its  rim 
bored  conical  has  a  special  rim  b  fastened  to  it.  The  pressure 
forcing  the  cone  a  into  the  cone  b  is  produced  by  a  series  of  springs 
in  place  of  a  single  central  spring  as  in  the  preceding  case.  A 


ART.  293] 


CONE  CLUTCHES 


411 


possible  advantage  of  this  arrangement  is  that  the  adjustment 
for  wear  may  be  made  more  easily;  also  the  pressure  may  be 
distributed  more  uniformly  over  the  surface  in  contact. 


FIG.  218. 

294.  Double -cone   Clutch. — The  clutches   described  in  Art. 
293  were  all  of  the  single-cone  type.     In  connection  with  hoist- 


412 


CONE  CLUTCHES 


[CHAP.  XVI 


ing  machinery,  machine  tools,  and  motor  cars,  it  is  not  unusual 
to  find  double-cone  clutches. 

(a)  Clyde  clutch. — The  design  of  a  double-cone  clutch  used  on 
hoisting   drums   manufactured   by   the   Clyde   Iron   Works   of 


FIG.  219. 


Duluth,  Minn.,  is  shown  in  detail  in  Fig.  223.  The  friction 
blocks  c  forming  one  member  of  the  clutch  are  fastened  to  the 
gear  b,  which  is  keyed  to  the  shaft  a.  The  clutch  is  engaged  by 
moving  the  drum  d  along  the  shaft  a  by  means  of  the  combination  of 


ART.  295] 


CONE  CLUTCH  ANALYSIS 


413 


lever  k,  screw  h,  thrust  pin  g,  cross-key  /,  and  collar  e.  A  spring 
I,  located  between  the  drum  and  the  gear,  automatically  disen- 
gages the  clutch  when  the  thrust  on  the  cross-key  is  released. 

In  place  of  using  a  double-cone  clutch,  several  manufacturers 
of  hoisting  drums  employ  one  of  the  single-cone  type,  operated 
practically  in  the  same  manner  as  explained  in  the  preceding 
paragraph.  The  Ingersoll  slip  gear,  described  in  Art.  231,  is 
nothing  more  than  a  form  of  double-cone  clutch. 

295.  Force  Analysis  of  a  Single-cone  Clutch. — In  the  follow- 
ing analysis  of  a  single-cone  clutch,  we  shall  assume  that  the 
outer  cone  is  the  driving  member  while  the  inner  cone  is  the 
member  having  an  axial  motion.  In  Fig. 
220  are  shown  the  various  forces  acting 
upon  the  inner  cone.  It  is  required  to 
determine  an  expression  for  the  moment 
M  that  the  clutch  is  capable  of  trans- 
mitting for  any  magnitude  of  the  axial 
force  P. 

Let    p  =  the  unit  normal  pressure  at 

the  surface  in  contact. 

r\  =  the  minimum  radius  of  the 

cone. 

r2  =  the  maximum  radius  of  the 

cone. 
n  =  the  coefficient  of  friction. 

The  maximum  moment  that  the  clutch 
will  transmit  is  equivalent  to  the  moment 
of  the  frictional  resistance  between  the 
inner  and  outer  cones.  The  normal  force  acting  upon  an  elemen- 


Fio.  220. 


tary  strip  of  the  surface  in  contact  is  2irrp 


dr 
sina 


The  com- 


ponent of  this  normal  pressure  parallel  to  the  line  of  action  of 
the  axial  force  P  is  2  irrpdrS  The  summation  of  these  compo- 
nents over  the  entire  surface  in  contact  must  equal  P;  hence 

P  =  2irfprdr  (420) 

The  force  of  friction  upon  the  elementary  strip  is  2ir^rp 


Bin  CK 


and  its  moment  about  the  axis  is 
moment  M  is  given  by 


sma 


Therefore  the 


414  CONE  CLUTCH  ANALYSIS  [CHAP.  XVI 


With  our  present  knowledge  of  friction,  it  is  impossible  to 
determine  a  correct  expression  for  the  moment  of  the  frictional 
resistance  between  the  two  elements  of  a  cone  clutch.  From  (421) 
it  is  evident  that  an  expression  for  the  moment  of  friction  de- 
pends upon  the  distribution  of  the  pressure  between  the  contact 
surfaces  as  well  as  the  variation  of  the  coefficient  of  friction. 
When  the  clutch  is  new  and  the  surfaces  are  machined  and  fitted 
correctly,  it  is  probable  that  the  pressure  is  nearly  uniformly 
distributed.  However,  after  the  clutch  has  been  in  service  for 
a  period  of  time,  there  will  be  a  redistribution  of  the  pressure 
due  to  the  unequal  wear  caused  by  the  different  velocities  along 
the  surfaces  in  contact.  This  variation  in  velocity  no  doubt 
results  in  a  change  in  the  value  of  the  coefficient  of  friction.  In 
view  of  the  fact  that  no  experimental  data  are  available,  we  shall 
assume  that  the  coefficient  of  friction  remains  constant,  and 
further,  that  the  normal  wear  at  any  point  is  proportional  to  the 
work  of  friction.  Denoting  the  normal  wear  at  any  point  by 
n,  the  law  just  stated  may  be  expressed  by  the  relation 

n  =  kpr  (422) 

Assuming  that  the  surfaces  in  contact  remain  conical,  it  follows 
that  the  normal  wear  is  constant;  hence 

p  =  ->  (423) 

in  which  C  denotes  the  ratio  of  the  constants  n  and  k.  Substi- 
tuting the  value  of  p  from  (423)  in  (420)  and  (421),  and  integrat- 
ing between  the  proper  limits,  we  obtain  the  following  relations: 

P  =  2  7rC(r2  -  n)  (424) 

M  =  ^-  (r\  -  rl)  (425) 

sin  a  v 

Eliminating  C  between  (424)  and  (425),  we  get  finally 


M  =  •-,  (426) 

2  sin  a 

in  -which  D  represents  the  mean  diameter  A  B  of  the  cone  shown 
in  Fig.  220. 


ART.  295]  CONE  CLUTCH  ANALYSIS  415 

To  determine  the  horse  power  that  a  cone  clutch  will  transmit, 
substitute  the  value  of  M  from  (426)  in  the  formula 

H  ==  63,030 
whence 


126,060  sin  a 
and  the  axial  force  is 


_  126,060  H  sin  a 


The  total  normal  pressure  is  given  by  the  following  expression : 

Pn  =  <^  fdr  =  ^-  (ra  -  r,) 
sin  ajri  sin  a 

Eliminating  C  by  means  of  (424),  it  is  evident  that 

sin  a 

The  total  normal  pressure  is  also  equal  to  the  average  intensity  of 
unit  normal  pressure  multiplied  by  the  total  area  in  contact;  or 

J—  =  irDfp'  (431) 

sin  a 

Combining  (428)  and  (431),  and  solving  for  H,  we  get 

vp'fND2 
"40^20" 

Denoting  the  product  of  ju  and  p'  by  the  symbol  K,  (432)  becomes 

(433) 


By  means  of  (433)  it  is  possible  to  determine  values  of  the  de- 
sign constant  K  for  cone  clutches  in  actual  service.  Such  values, 
if  based  on  clutches  in  successful  operation,  will  prove  of  consider- 
able help  in  the  design  of  new  clutches.  The  analysis  used  in 
deriving  (433)  is  similar  to  that  first  proposed  by  Mr.  John  Edgar 
in  the  American  Machinist  of  June  29,  1905,  though  he  applied 
his  formulas  to  expanding  ring  clutches. 

Another  design  constant  that  may  be  found  useful  in  arriving 
at  the  proportions  of  a  cone  clutch  is  that  which  represents  the 
number  of  foot-pounds  of  energy  per  minute  that  can  be  trans- 


416  STUDY  OF  CONE  CLUTCHES  [CHAP.  XVI 

mitted  per  square  inch  of  contact  surface  of  the  clutch.     Denot- 
ing this  constant  by  the  symbol  K\y  we  find  that 

„.        10,500  H 

Ki  =  (434) 

296.  A  Study  of  Cone  Clutches. — Through  the  generosity  and 
cooperation  of  about  forty  automobile  manufacturers,  informa- 
tion pertaining  to  a  large  number  of  cone  clutches  was  obtained. 
Some  of  the  clutches  that  were  analyzed  were  faced  with  leather, 
others  with  asbestos  fabric,  and  a  few  were  equipped  with  cork 
inserts. 

From  the  information  furnished  by  the  various  manufacturers, 
it  was  possible  to  determine  for  each  clutch  the  magnitude  of  the 
design  constant  K  and  the  intensity  of  the  unit  normal  pressure 
pf.  With  K  and  pf  known,  the  probable  value  of  the  coefficient 
of  friction  fj,  was  calculated  The  values  of  K,  p',  and  /x  were 
found  to  vary  with  the  mean  velocity  of  the  surface  in  contact. 
In  this,  as  well  as  in  all  other  analyses  of  motor-car  clutches,  the 
values  of  K  are  based  upon  the  horse  power  and  speed  correspond- 
ing to  the  maximum  torque  of  the  motor,  and  not  upon  the  maxi- 
mum horse  power  transmitted  and  the  speed  corresponding 
thereto.  It  should  be  remembered  that  clutches  must  be  de- 
signed for  the  maximum  loads  coming  upon  them,  and  in  the 
case  of  motor  cars,  the  loads  are  greatest  when  the  motor  trans- 
mits the  maximum  torque. 

(a)  Leather-faced  cone  clutches. — For  the  leather-faced  cone 
clutches  analyzed,  the  values  of  K  and  pf  were  plotted  on  a  speed 
base,  and  the  curves  shown  in  Fig.  221  represent  the  average 
results.  From  this  figure,  it  is  apparent  that  the  magnitude  of 
K  decreases  with  an  increase  in  the  mean  velocity  of  the  sur- 
faces in  contact.  The  intensity  of  the  unit  normal  pressure  pf 
also  decreases  with  an  increase  in  the  velocity.  The  value  of  the 
coefficient  of  friction  was  also  plotted  on  a  speed  base,  and  an 
average  curve  passed  through  the  series  of  points.  For  all  prac- 
tical purposes,  the  average  value  of  AI  may  be  represented  by  a 
straight  line  parallel  to  the  velocity  axis,  giving  a  constant  value 
of  fj,  equal  to  0.2  for  all  speeds.  The  jj.  curve  is  not  shown  in 
Fig.  221. 

(6)  Cone  clutches  faced  with  asbestos  fabric. — At  the  present 
time  there  are  only  a  few  motor-car  builders  using  cone  clutches 
faced  with  asbestos  fabric.  From  an  investigation  of  six  such 


Art.  296] 


STUDY  OF  CONE  CLUTCHES 


417 


clutches,  K  was  found  to  vary  from  1.95  to  4.77.  The  intensity 
of  the  unit  normal  pressure  pf  varied  from  9.5  to  17  pounds  per 
square  inch.  Until  such  a  time  as  more  information  pertaining 
to  asbestos  fabric  facing  is  available,  it  is  suggested  that  the 
values  of  K  given  in  Fig.  221  be  used,  and  that  the  coefficient  of 
friction  be  assumed  as  0.30. 

(c)  Cone  clutches  with  cork  inserts. — It  was  impossible  to  get 
information  pertaining  to  a  large  number  of  cone  clutches  having 
cork  inserts,  since  very  few  motor-car  builders  are  using  them  at 
the  present  time.  Four  such  clutches  were  analyzed  and  the 


2000      2500     3000      3500     4000 
Velocity  in  ft.  per  min. 
FIG.  221. 


4500 


5000 


design  constant  K  was  found  to  vary  from  2.2  to  3.1.  Until  such 
a  time  as  sufficient  information  is  available  for  a  more  extended 
analysis,  it  would  seem  advisable  to  use  the  values  of  K  given 
for  the  leather-faced  cones  when  making  calculations  for  cork 
insert  clutches.  The  coefficient  of  friction  may  be  assumed  as 
0.25. 

(d)  Cone-face  angle. — In  the  study  of  the  motor-car  cone 
clutches,  it  was  found  that  for  a  leather  facing  the  face  angle  a 
varied  from  10  to  13  degrees.  The  majority  of  the  manufac- 
turers are  using  12^  degrees  which  is  now  recommended  as  a  stand- 
ard by  the  Society  of  Automotive  Engineers.  With  an  asbestos- 


418 


CONE  CLUTCH  INVESTIGATION 


[CHAP.  XVI 
degrees,  and  for 


fabric  facing,  the  angle  a  varied  from  11  to  14 
the  cork  insert  clutches,  from  8  to  12  degrees. 

297.  Experimental  Investigation  of  a  Cone  Clutch. — In  the 

Zeitschrift  des  Vereines  deutscher  Ingenieure,  for  Dec.  15,  1915, 
Prof.  H.  Bonte  of  Karlsruhe  presented  an  article  in  .which  he  gave 
the  results  of  an  experimental  investigation  of  a  cone  clutch. 
The  two  halves  of  the  clutch  were  made  of  cast  iron,  and  during 


600 

500 
.a 

c 
-a    400 

x 

^- 

/ 

/ 

/ 

0> 

f 

'  — 

/ 

^ 

£ 

s 

£    300 

^/ 

7^ 

\ 

,L- 

r/ 

H 

\p- 

—^ 

*  — 

// 

f 

-f 
c    ™« 

/ 

/ 

u    200 

Q 

/ 

^ 

z 

/ 

XV 

roo 

1  0J 

50 

100 
Axial 

J50             200 

Pre  s  sure      in 
FIG.  222. 

250 
Ib. 

300 

35< 

the  test  the  surfaces  in  contact  were  lubricated.  The  main  ob- 
ject in  undertaking  the  investigation  was  to  determine  which  of 
the  following  two  formulas,  generally  quoted  in  technical  works, 
was  the  correct  one  to  use  in  designing  cone  clutches : 


M  = 


2  sin 


or 


M 


2  (sin  a  +  ju  cos  a) 


(435) 


(436) 


ART.  297]  CONE  CLUTCH  INVESTIGATION  419 

In  Fig.  222  are  plotted  the  results  of  Prof.  Bonte's  experimental 
investigation  on  a  clutch  having  an  angle  a  =  15  degrees.  In 
this  figure  are  included  the  results  obtained  by  evaluating  (435) 
and  (436).  The  results  obtained  by  the  use  of  (435)  are  rep- 
resented by  the  dot  and  dash  line,  and  those  obtained  by  the 
use  of  (436)  are  represented  by  the  dash  line.  The  following 
conclusions  may  be  arrived  at  from  the  results  published  by 
Bonte. 

(a)  When  the  angle  a  is  15  degrees,  the  error  introduced  by 
using  (436)  is  large,  while  the  agreement  between  the  experi- 
mental results  and  those  obtained  by  using  (435)  is  very  close. 

(b)  When  the  angle  a  is  30  degrees,  the  experimental  results 
lie  between  those  obtained  by  (435)  and  (436).     The  curve  repre- 
senting the  results  obtained  by  (435)  lies  above,  but  is  much 
closer  to  the  experimental  curve  than  that  obtained  by  (436). 

(c)  For  the  angle  45  degrees  and  60  degrees,  the  experimental 
points  lay  above  those  obtained  by  (435),  which  in  turn  lay 
above  the  points  obtained  by  (436). 

(d)  Apparently,  the  coefficient  of  friction  is  not  constant  as 
generally  assumed  but  varies  slightly  with  the  pressure. 

As  a  result  of  this  experimental  investigation,  Prof.  Bonte 
makes  a  plea  that  (436),  which  is  apparently  incorrect,  should 
no  longer  be  used  in  designing  cone  clutches. 

298.  Analysis  of  a  Double-cone  Clutch. — For  the  double-cone 
clutch  shown  in  Fig.  223,  it  is  required  to  determine  an  expres- 
sion for  the  force  F  that  must  be  applied  at  the  end  of  the  lever 
k  in  order  to  engage  the  clutch;  also,  to  determine  the  maximum 
moment  that  the  clutch  is  capable  of  transmitting. 
Let  DI  =  the  mean  diameter  of  the  smaller  cone. 
D2  =  the  mean  diameter  of  the  larger  cone. 
D3  =  the  mean  diameter  of  the  thrust  collar  e. 
Z)4  =  the  mean  diameter  of  the  spring  cage  m. 
L  =  the  length  of  the  lever  arm  k. 

P  =  the  axial  force  holding  the  drum  against  the  V  blocks. 
S  =  the  spring  force. 
d  =  the  mean  diameter  of  the  screw  h. 
ft  =  the  angle  of  rise  of  the  mean  helix  of  the  screw. 
<p'  =  the  angle  of  friction  for  the  screw. 
Mo  =  the   coefficient  of  friction  between  the  drum  and 

blocks  c. 

jit  =  the  coefficient  of  friction  between  the  drum  and 
collar  e  and  cage  m. 


420 


DOUBLE-CONE  CLUTCH  ANALYSIS       [CHAP.  XVI 


Consulting  Fig.  223,  it  is  evident  that  the  axial  pressure  that 
the  screw  h  must  produce  is  P  +  S  plus  the  force  required  to 
move  the  drum  with  its  load  along  the  shaft.  The  latter  force  is 
relatively  small  and  may  be  accounted  for  by  considering  it 
equivalent  to  a  certain  percentage  of  the  total  pressure  produced. 
Calling  Q  the  total  pressure  due  to  the  screw,  we  find  that  its 
magnitude  may  be  expressed  by  the  formula 


P  +  S 


(437) 


in    which  77  may  be  assumed  as  equal  to  0.97.     From  this  it 
follows  that  the  force  F  required  on  the  operating  lever  k,  in 


\JU 


FIG.  223. 


V 

order  to  produce  the  axial  pressure  Q,  must  have  a  magnitude 
given  by  the  expression 


F  = 


tan  05 +  ,0 


(438) 


The  total  moment  that  the  drum  will  transmit  is  equivalent 
to  the  sum  of  the  moments  of  friction  of  the  double  cone  c,  of 
the  thrust  collar  e,  and  of  the  spring  cage  m.  The  last  two 
moments  just  mentioned  are  usually  small  when  compared  with 
the  first,  and  frequently  are  not  considered  at  all.  The  moment 


ART.  298 j 


DOUBLE-CONE  CLUTCH  ANALYSIS 


421 


transmitted  by  the  double  cone  is  equivalent  to  the  sum  of  the 
moments  of  the  two  cones  taken  separately,  or 


M  2  = 


(439) 


4  sin  a  v 

The  sum  of  the  moments  transmitted  by  the  collar  e  and  the 
spring  cage  m  is 

M3  +  M<  =  ^(P  +  S)+  ^  (440) 

Adding  (439)  and  (440) ,  we  find  that  the  total  moment  trans- 
mitted by  the  drum  has  the  magnitude 

M  =  Ml  +  M2  +  M3  +  Mi  (441) 

299.  Smoothness    of    Engagement    of    Cone    Clutches. — In 

motor-car  service,  it  is  very  desirable  that  the  car  be  started 


FIG.  224. 

without  jerks.  In  order  to  secure  smooth  clutch  engagement, 
the  designers  of  clutches  were  compelled  to  originate  devices 
that  insured  evenness  of  contact  between  the  friction  surfaces. 
A  few  such  devices,  as  applied  to  cone  clutches,  are  shown  in 
Figs.  224  to  227,  inclusive.  In  general,  it  may  be  said  that  the 
function  of  these  devices  is  to  raise  slightly  the  cone  facing 
at  intervals  around  the  periphery,  so  that  upon  engagement 
only  a  small  portion  of  the  friction  surface  comes  into  contact 
with  the  flywheel  rim.  As  soon  as  the  full  spring  pressure 
is  exerted,  the  facing  is  depressed  and  the  entire  surface  of 
the  cone  becomes  effective.  One  disadvantage  of  the  attach- 


422 


CONE  CLUTCH  ENGAGING  DEVICE       [CHAP.  XVI 


ments  just  discussed  is  that  they  tend  to  increase  the  spinning 
effect  due  to  the  extra  weight  added  to  the  periphery  of  the  cone. 


(a) 


FIG.  225. 


(a) 


(b) 


FIQ.  226. 


(a) 


(b) 


FIG.  227. 


A  few  manufacturers  are  using  cork  inserts  in   connection 
with  their  leather-faced  cone  clutches.     It  is  claimed  that  in 


ART.  301]  SINGLE  DISC  CLUTCHES  423 

addition  to  increasing  the  coefficient  of  friction  between  the 
surfaces  in  contact,  the  cork  inserts  have  the  effect  of  producing 
smooth  and  easy  engagement  of  the  clutch.  Obviously,  cork 
inserts  have  another  advantage  in  that  the  weight  of  the  cone 
is  actually  decreased,  thereby  decreasing  the  spinning  effect. 
Fig.  227(6)  shows  one  method  of  holding  cork  inserts  in  the  facing 
of  a  cone  clutch. 

300.  Clutch  Brakes. — In  addition  to  securing  smooth  and  easy 
clutch  engagement,  some  means  must  be  provided  to  prevent 
the  " spinning"  of  the  clutch  when  it  is  disengaged.     By  keeping 
the  size  and  weight  of  the  clutch  down  to  a  minimum,  spinning 
may  be  reduced  slightly.     However,  to  overcome  the  spinning 
action  completely,  small  brakes  that  are  brought  into  action  when 
the  pedal  is  depressed  must  be  provided      A  cone  clutch  equipped 
with  such  an  auxiliary  brake  is  shown  in  Fig.  218,  and  in  Figs. 
229,  236,  and  241  are  shown  disc  clutches  equipped  with  such 
brakes. 

DISC  CLUTCHES 

In  general,  a  disc  clutch  consists  of  a  series  of  discs  arranged 
in  such  a  manner  that  each  driven  disc  is  located  between  two 
driving  discs.  Disc  clutches  are  made  in  various  forms,  as  a 
study  of  the  designs  used  in  connection  with  various  classes  of 
machinery  will  show.  For  convenience,  disc  clutches  will  be 
classified  as  follows: 

(a)  Single-disc  type,  in  which  a  single  disc  serves  as  the  driven 
member. 

(6)  Multiple-disc  type,  in  which  two  or  more  discs  act  as  the 
driven  member. 

301.  Single-disc  Clutch.— In  Figs.  228  to  233,  inclusive,  are 
shown  six  designs  of  single-disc  clutches,  the  first  two  represent- 
ing the  practice  of  two  motor-car  builders,  and  the  third  and 
fourth  showing  the  details  of  two  clutches  used  for  general  power- 
transmission    purposes.     The    remaining    two,    namely,    those 
shown  in  Figs.  232  and  233,  are  intended  for  special  purposes. 
As  in  the  case  of  the  cone  clutches,  the  development  of  the  auto- 
mobile is  responsible  to  a  large  extent  for  the  advances  made  in 
the  design  of  disc  clutches. 

(a)  Knox  clutch. — The  disc  clutch  shown  in  Fig.  228  is  that 
used  on  the  old  Knox  motor  cars.  The  discs  a  and  b  are  fastened 


424 


KNOX  CLUTCH 


[CHAP.  XVI 


to  the  flywheel  while  the  driven  disc  c  is  fastened  to  the  flange  d, 
which  in  turn  is  splined  to  the  transmission  shaft  e.  Due  to  the 
action  of  a  series  of  springs  located  in  the  rim  of  the  flywheel,  the 
driven  disc  c  is  clamped  between  the  two  driving  discs.  The 
clutch  is  released  by  overcoming  the  spring  force  upon  the  discs 
b,  through  the  medium  of  the  sliding  sleeve  /,  lever  g,  and  plunger 


h.  All  of  the  discs  used  in  this  clutch  are  made  of  cast  iron.  In 
order  to  obtain  smooth  engagement  and  to  increase  the  coefficient 
of  friction  between  the  surfaces  in  contact,  the  driven  disc  c  is 
fitted  with  cork  inserts  as  shown. 

(b)   Velie  clutch. — The  type  of  single-disc  clutch  used  on  the 
Velie  motor  cars  is  shown  in  Fig.  229.     Instead  of  having  two 


ART.  301] 


VELIE  CLUTCH 


425 


driving  discs  as  in  the  Knox  clutch,  this  design  has  only  one 
driving  disc  b,  but  the  web  of  the  flywheel  serves  the  same  pur- 
pose as  a  second  disc.  The  steel  driven  disc  c  is  riveted  to  the 
flange  of  the  clutch  drive  shaft  d.  The  clutch  is  kept  in  engage- 
ment by  the  conical  spiral  spring  pressing  upon  a  bronze  sleeve, 
which  in  turn  transmits  the  pressure  to  the  wedge  /  by  means  of 


FIG.  229. 

suitable  links.  The  back  face  of  the  driving  disc  6,  as  well  as 
the  inside  face  of  the  cover  plate  a,  is  bored  conical  to  fit  the 
wedge  /.  The  cover  plate  screws  into  the  flywheel  and  is  locked 
to  it  by  means  of  the  set  screws  shown.  To  release  the  clutch, 
the  wedge  is  withdrawn  slightly  by  forcing  the  bronze  sleeve 
back  against  the  action  of  the  spring.  The  treadle  operates  the 
releasing  collar  g  by  means  of  a  system  of  links  and  levers.  In 


426 


PLAMONDON  CLUTCH 


[CHAP.  XVI 


the  Velie  clutch,  the  driven  disc  c  is  faced  on  both  sides  with  an 
asbestos  fabric,  called  Raybestos.  Attention  is  directed  to  the 
small  disc  brake  which  prevents  excessive  spinning  when  the 
clutch  is  released. 

(c)  Plamondon  dutch. — A  sectional  view  of  the  Plamondon  disc 
clutch  as  applied  to  a  pulley  running  loose  on  a  shaft  a,  is  shown 
in  Fig.  230.  The  disc  c,  which  is  faced  with  hard  maple  seg- 
ments, is  made  in  halves  so  that  it  can  be  removed  in  case  the 
friction  blocks  require  renewal.  The  flange  d  slides  on  the  flanged 
hub  e,  which  is  keyed  to  the  shaft  a.  By  means  of  the  compound 


FIG.  230. 

toggle  levers,  /,  g,  and  h,  the  flanges  d  and  e  are  pressed  against 
the  disc  c,  thereby  transmitting  the  power  from  the  pulley  to  the 
shaft,  or  vice  versa.  Attention  is  called  to  the  simplicity  of  this 
clutch  and  also  to  the  ease  with  which  adjustments  for  wear  may 
be  made. 

(d)  E.  G.  I.  clutch. — In  Fig.  231  is  shown  another  design  of  a 
single-disc  clutch,  but  in  this  case  the  pressure  upon  the  discs  is 
produced  by  a  system  of  rollers  and  levers  instead  of  springs  or 
toggle  joints.  The  cast-iron  discs  c  and  d  are  made  to  rotate 


ART.  301] 


E.  G.  I.  CLUTCH 


427 


with  the  casing  b  by  means  of  the  three  bolts  e.  The  casing  b  is 
fastened  to  the  shaft  a  by  set  screws  or  keys.  Between  the  slid- 
ing discs  c  and  d  is  located  a  third  disc  I,  to  the  hub  of  which  may 
be  fastened  a  gear  or  pulley.  The  pressure  exerted  by  the  sliding 
discs  upon  the  disc  I  is  produced  by  shifting  the  sleeve /in  ward. 
This  movement  causes  the  levers  h  to  assume  a  position  perpen- 
dicular to  the  shaft,  thereby  forcing  apart  the  disc  c  and  the 
casing  6,  and  at  the  same  time  creating  a  considerable  pressure 
upon  the  disc  I.  Upon  disengagement  of  the  clutch,  the  springs 


FIG.  231. 

m  spread  the  discs  c  and  d.     The  disc  I  is  fitted  with  a  series  of 
wooden  plugs,  as  shown  in  the  figure. 

302.  Hydraulically  Operated  Disc  Clutch. — In  such  naval  ves- 
sels as  torpedo  boat  destroyers,  it  has  been  found  that  a  combina- 
tion of  reciprocating  engines  with  turbines  gives  better  economy 
over  a  wide  range  of  speed  than  turbines  alone.  The  engines  are 
used  for  cruising  speeds  only,  and  exhaust  into  the  low-pressure 
turbines.  At  the  higher  speeds,  the  ship  is  propelled  by  turbines 
only.  According  to  the  machinery  specifications  drawn  up  by 


428 


METTEN  HYDRAULIC  CLUTCH 


[CHAP.  XVI 


the  Navy  Department  for  some  of  the  latest  types  of  destroyers, 
the  installation  of  turbines  and  cruising  engines  called  for  must 
fulfill  the  following  conditions: 

(a)  That  the  engines  and  turbines  should  be  capable  of  oper- 
ating in  combination  on  cruising  speed. 

(6)  That  the  turbines  should  be  capable  of  operating  alone,  the 

engines  standing  idle. 

(c)  That  means  should  be 
provided  whereby  the  cruising 
engines  may  be  connected  to 
or  disconnected  from  the  tur- 
bine shafts  without  stopping 
the  propelling  machinery. 

It  is  evident  from  the  above 
specifications  that  some  form 
of  reliable  clutch  is  necessary 
to  fulfill  condition  (c) ,  and  in 
order  to  meet  this  require- 
ment, Mr.  J.  F.  Metten,  Chief 
Engineer  of  the  Wm.  Cramp 
and  Sons  Ship  and  Engine 
Building  Co.,  developed  and 
patented  the  single-disc  clutch 
shown  in  Fig.  232.  The  hol- 
low crankshaft  a  of  the  re- 
ciprocating engine  has  con- 
nected to  it  the  head  6,  which 
in  combination  with  the  steel 
frame  c  forms  the  driving 
member  of  the  clutch.  The 
inner  face  of  the  frame  c  is 
lined  with  an  asbestos  fabric. 
„  Inside  of  this  driving  member 

.t  IG.    jL6£. 

and  attached  to  it,  is  located 

a  movable  member  consisting  of  the  spherical  steel-plate  head  e, 
ring  /  faced  with  asbestos  fabric,  and  the  flexible  ring  g.  The 
shaft  I,  which  is  an  extension  of  the  main  turbine  shaft,  has  bolted 
to  its  flange  a  steel-plate  disc  k,  Y±  inch  thick.  When  oil  under 
pressure  is  forced  through  the  hollow  crankshaft  into  the  pressure 
chamber  formed  between  the  heads  b  and  e  and  rings  /  and  g,  the 
disc  k  is  gripped  by  the  friction  surfaces  d  and  h.  As  shown  in 


ART.  302]  METTEN  HYDRAULIC  CLUTCH  429 

Fig.  232,  the  clutch  is  disengaged.  In  order  to  insure  quick  dis- 
engagement of  the  clutch,  the  flexible  ring  g  is  so  designed  that 
its  contraction  upon  release  of  the  oil  pressure  will  force  the  oil 
out  of  the  pressure  chamber. 

The  axial  force  available  for  creating  the  frictional  resistance 
on  the  disc  k  is  that  due  to  the  fluid  pressure  upon  the  combined 
unbalanced  areas  of  the  head  e  and  ring  /,  minus  the  resistance 
that  the  flexible  ring  g  offers  to  extension.  Having  determined 
the  axial  force,  and  knowing  the  inner  and  outer  diameters  of  the 
contact  surfaces  d  and  h,  the  probable  horse  power  that  the  clutch 
is  capable  of  transmitting  may  readily  be  determined. 

303.  Slip  Coupling. — In  many  installations,  it  is  desirable  to 
place  between  a  motor  and  the  driven  machine  or  mechanism 
some  form  of  coupling  that  will  slip  when  the  load  is  excessive, 
thus  protecting  the  motor  against  overloads.     The  details  of  such 
a  coupling,  called  a  slip  coupling,  are  shown  in  Fig.  233,  which 
represents  the  design  used  by  the  Illinois  Steel  Co.  and  others 
on  the  drives  of  rolling  mill  tables.     A  modification  of  this  design 
is  also  used  on  the  furnace-charging  machines  found  in  steel 
works.     The  slip  coupling  illustrated  in  Fig.  233  is  nothing  more 
than  a  single-disc  friction  coupling.     The  flanged  hub  a  is  keyed 
to  the  driving  shaft,  and  has  bolted  to  its  rim  a  plate  b.     Be- 
tween a  and  &,  and  separated  from  them  by  fiber  discs,  is  the 
flanged  hub  c  which  is  keyed  to  the  driven  shaft.     The  bolts  con- 
necting the  plate  b  with  the  hub  a  are  provided  with  springs 
which  create  a  pressure  on  both  faces  of  the  hub  c.     The  torsional 
moment  transmitted  by  the  coupling  depends  directly  upon  this 
spring  pressure,  which  may  be  varied  by  merely  adjusting  the 
nuts  of  the  coupling  bolts.     In  Table  93   are  given  the  gen- 
eral proportions  of  a  series  of  sizes  of  the  slip  coupling  shown  in 
Fig.  233,  and  these  proportions  represent  the  practice  of  the 
Illinois  Steel  Co. 

304.  Multiple-disc  Clutches. — In  Figs.  234  to  237,  inclusive, 
are  shown  four  designs  of  multiple-disc  clutches,  the  first  two  of 
which  represent  the  practice  of  two  manufacturers  of  transmis- 
sion clutches,  and  the  last  two  show  the  type  of  multiple  disc 
clutches  used  on  motor  cars. 

(a)  Akron  clutch. — The  Akron  clutch  shown  in  Fig.  234  is  a 
double-disc  clutch  employing  an  ingenious  roller  toggle  for  pro- 
ducing the  pressure  between  the  discs.  The  clutch  consists  of 


430 


SLIP  COUPLING 


[CHAP.  XVI 


a  casing  a  upon  the  hub  of  which  gears,  sprockets,  or  pulleys  may 
be  keyed.  Into  the  casing  a  is  screwed  a  head  b  having  a  series 
of  notches  on  its  periphery,  into  which  the  locking  set  screw  c 
projects.  This  combination  of  screwed  head  and  set  screws 
affords  a  simple  and  effective  means  of  making  adjustments  for 
wear.  The  inner  face  of  the  head  b  and  that  of  the  casing  a  are 


machined  and  serve  as  contact  surfaces  for  the  discs  d  and  e, 
respectively.  The  discs  are  splined  to  the  hub  /,  which  in  turn 
is  keyed  to  the  shaft  g.  To  engage  the  clutch,  the  sliding  sleeve 
h  is  moved  outward,  thus  pulling  the  forked  levers  k  with  it, 
and  as  a  result  of  the  action  of  the  roller  toggle,  forcing  the  discs 


ART.  303] 


SLIP  COUPLING 


431 


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432 


AKRON  CLUTCH 


[CHAP.  XVI 


d  and  e  apart.  The  clutch  is  lubricated  effectively  by  having 
the  casing  partially  filled  with  oil,  and  hence  the  wear  on  the 
friction  surfaces  is  reduced  to  a  minimum. 


FIG.  234. 


(b)  Dodge  clutch. — The  multiple-disc  clutch  shown  in  Fig. 
235  is  used  for  general  transmission  service.  The  cylindrical 
casing  c  with  its  hub  b  is  keyed  to  the  shaft  a,  and  may  serve 


FIG.  235. 


either  as  the  driving  or  the  driven  member.  The  discs  e,  fitted 
with  wood  blocks,  rotate  with  c  and  at  the  same  time  may  be 
moved  in  an  axial  direction.  The  flanged  hub  /  is  keyed  to  the 


ART.  304] 


ALCO  CLUTCH 


433 


shaft  k  and  has  splined  to  it  the  two  discs  g  and  h,  the  outer  one 
of  which  may  be  moved  forward  by  the  roller  toggle  operating 
mechanism.  The  axial  movement  given  to  h  clamps  the  various 
discs  together,  thus  transmitting  the  desired  power.  It  should 
be  noted  that  means  for  taking  up  wear  on  the  discs  are  provided, 
and  that  the  clutch  is  self-lubricating.  An  oil  ring  m  revolves 


FIG.  236. 


upon  the  shaft  and  carries  a  continuous  supply  of  lubricant 
from  the  oil  reservoir  below  to  all  parts  of  the  sleeve. 

(c)  Alco  clutch. — The  multiple-disc  clutch  used  on  the  Alco 
car,  formerly  manufactured  by  the  American  Locomotive  Co., 
is  shown  in  Fig.  236.  As  shown  in  the  figure,  the  driving  discs 
are  connected  to  the  flywheel  through  the  hollow  pin  b  and  the 


434 


PATHFINDER  CLUTCH 


[CHAP.  XVI 


drum  a,  while  the  driven  discs  are  splined  to  the  inner  hub  c 
which  is  keyed  to  the  clutch  shaft  d.  Both  driving  and  driven 
discs  are  so  mounted  that  they  must  rotate  with  the  member  to 
which  they  are  connected,  and  at  the  same  time  these  discs  may 
move  in  an  axial  direction.  To  disengage  the  clutch,  the  collar 
e  is  moved  to  the  right  carrying  with  it  the  sleeve  /  and  the  spider 
g,  thus  releasing  the  pressure  between  the  two  sets  of  discs.  As 
soon  as  the  treadle  is  released,  the  spring  will  engage  the  clutch. 

Both  sides  of  the  driving 
discs  are  faced  with 
Raybestos. 

(d)  Pathfinder  clutch. 
— Another  form  of  multi- 
ple-disc clutch  is  shown 
in  Fig.  237,  and  as  in  the 
Alco  clutch,  both  sides 
of  the  driving  discs  are 
faced  with  asbestos 
fabric.  The  latter 
clutch  is  much  shorter 
in  length  than  the 
former,  and  in  place  of  a 
single  spring  to  create 
the  axial  pressure  upon 
the  discs,  a  double  con- 
centric spring  is  used. 
The  pressure  upon  the 
treadle  is  transmitted  to 
the  collar  e  on  the  trans-* 
mission  shaft  d  by  the 
shipper  arm  k,  through 
FIG.  237.  the  medium  of  a  ball 

bearing  m,  as  shown  in 

the  figure.  In  general,  the  description  and  operation  of  the  Path- 
finder clutch  is  similar  to  that  of  the  Alco  clutch. 

305.  Force  Analysis  of  a  Disc  Clutch. — It  is  required  to  de- 
termine an  expression  for  the  moment  M  that  the  clutch  is 
capable  of  transmitting  for  a  given  magnitude  of  the  axial  force 
P.  We  shall  assume,  in  the  following  analysis,  that  the  law 
expressed  by  (422)  will  hold  for  disc  clutches.  This  is  approxi- 
mately true,  especially  for  clutches  having  very  narrow  contact 
surfaces. 


ART.  305]  DISC  CLUTCH  ANALYSIS  435 

Let  D  =  the  mean  diameter  of  the  discs. 
TI  =  the  minimum  radius  of  the  discs. 
r2  =•  the  maximum  radius  of  the  discs. 
s  =  the  number  of  friction  surfaces  transmitting  power. 

The  general  expressions  deduced  for  the  conical  clutch  may 
be  applied  to  the  disc  clutch  by  making  the  angle  a  =  90  de- 
grees. Substituting  this  value  in  (426),  we  get  for  the  moment 
for  each  contact  surface 


l 

2 

Hence  for  s  surfaces,  the  total  moment  becomes 

M=^  (442) 

Substituting  (442)  in  (427),  we  find  that  the  horse  power  trans- 
mitted by  a  disc  clutch  is  given  by  the  expression 

_  -B.PDN 
~  12p60 
from  which  the  axial  force  is 


The  total  axial  pressure  P  is  also  given  by  the  product  of  the 
area  of  contact  of  one  disc  and  the  average  intensity  of  normal 
pressure  p',  that  is 

P  =  '  (rl-rl)  P'  (445) 

Combining  (444)  and  (445),  and  solving  for  H,  we  obtain  the 
following  expression 


in  which  /  denotes  the  face  of  the  contact  surface,  or  (r2  —  n). 
Replacing  up'  by  the  symbol  K2,  as  was  done  in  Art.  295,  (446) 
becomes 

* 


40,120 

In  disc  clutches,  as  in  cone  clutches,  it  may  be  desirable  to 
know  the  number  of  foot-pounds  of  energy  the  clutch  will  trans- 


436 


STUDY  OF  DISC  CLUTCHES 


[CHAP.  XVI 


mit  per  minute  per  square  inch  of  actual  contact  surface.     Repre- 
senting this  factor  by  the  symbol  K3)  we  get 

10,500  H 


sfD 


(448) 


306.  A  Study  of  Disc  Clutches. — (a)  Motor-car  clutches  — 
A  study  of  disc  clutches  used  on  motor  cars  discloses  the  fact  that 
the  majority  of  such  clutches  have  steel  discs  in  contact  with 
asbestos-fabric-faced  steel  discs.  Among  other  combinations 
that  are  used  for  the  friction  surfaces,  the  following  may  be 


20 


0> 

I I0 


0.2 


Jo, 


1000 


1500 


2000 


2500 


3000 


3500 


Mean     Velocity  -  -Ft.    per    min. 
FIG.  238. 


mentioned:  (1)  steel  against  steel;  (2)  steel  against  steel  with 
cork  inserts;  (3)  steel  against  bronze. 

An  analysis,  similar  to  that  of  cone  clutches,  was  made  of  a 
large  number  of  different  types  of  disc  clutches  used  on  motor 
cars.  The  information  required  for  such  an  analysis  was  fur- 
nished by  the  various  motor-car  manufacturers.  The  graphs 
plotted  in  Fig.  238  represent  the  average  results  obtained  for 
the  asbestos-fabric-faced  disc  clutches  running  dry,  and  are 
based  upon  an  investigation  of  at  least  thirty-five  different 
clutches.  The  values  of  K%  were  obtained  by  evaluating  (447), 


ART.  306] 


STUDY  OF  DISC  CLUTCHES 


437 


while  those  of  p'  were  deter- 
mined by  means  of  (445). 
The  graph  for  the  coefficient  of 
friction  /*  was  established  from 
the  relation  K<z  =  up'. 

For  clutches  employing  the 
other  friction  surfaces  men- 
tioned in  a  preceding  paragraph, 
it  was  thought  best  not  to  rep- 
resent the  results  graphically, 
since  there  was  not  sufficient  in- 
formation available  to  warrant 
definite  conclusions.  However, 
in  Table  94  are  exhibited  the 
minimum,  maximum,  and  mean 
values  of  the  design  constant 
K2,  of  the  average  intensity  of 
normal  pressure  p',  and  of  the 
coefficient  of  friction  for  the 
various  types  of  motor-car  disc 
clutches  investigated. 

(b)  Transmission  clutches. — 
Through  the  generosity  of 
several  manufacturers  of  trans- 
mission clutches,  considerable 
information  was  obtained  which 
made  it  possible  to  carry  out  an 
analysis  similar  to  that  on 
motor-car  clutches  mentioned 
above.  Since  no  information 
regarding  the  axial  pressure 
upon  the  discs  was  available,  it 
was  impossible  to  determine  the 
probable  values  of  p'  and  /*,  and 
consequently  only  the  relation 
between  the  design  constant  K2 
and  the  mean  velocity  of  the 
friction  surfaces  at  100  revolu- 
tions per  minute  of  the  clutch 
was  calculated.  The  reason 
for  selecting  the  mean  velocity 


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438 


STUDY  OF  DISC  CLUTCHES 


[CHAP.  XVI 


at  100  revolutions  per  minute  of  the  clutch  as  one  of  the  vari- 
ables is  the  fact  that  all  of  the  manufacturers  rate  their  clutches 
at  this  speed. 

The  disc  clutches  investigated  were  fitted  with  the  following 
combinations  of  friction  surfaces:  cast  iron  against  wood;  cast 
iron  against  compressed  paper  and  wood;  cast  iron  against  cast 
iron;  cast  iron  against  cast  iron  with  cork  inserts. 

1.  For  clutches  equipped  with  cast-iron  discs  in  contact  with 
wood-faced  discs,  it  was  found  that  the  design  constant  K2 


25 


20 


15 


10 


100  200  500  400 

Mean    Velocity  -  f-h    per    min. 
FIG.  239. 

varied  between  wide  limits.  This  variation  is  clearly  shown  in 
Fig.  239,  in  which  the  two  curves  represent  the  maximum  and 
minimum  results  obtained. 

2.  For  clutches  having  cast-iron  discs  in  contact  with  discs 
faced  with  compressed  paper,  the  relation  existing  between  K2 
and  the  mean  velocity  at  100  revolutions  per  minute  of  the 
clutch  is  represented  by  the  graph  of  Fig.  240. 

3.  For  clutches  having  cast-iron  friction  surfaces,  the  relation 
between  the  design  constant  K%  and  the  mean  velocity  may  be 
expressed  by  the  following  formula: 

K2  =  18  -  ~>  (449) 


ART  307] 


HELE-SHAW  CLUTCH 


439 


in  which  V  denotes  the  mean  velocity  of  the  friction  surfaces 
at  100  revolutions  per  minute  of  the  clutch. 

4.  For  clutches  having  cast-iron  discs  in  contact  with  cast- 
iron  discs  fitted  with  cork  inserts,  the  relation  between  K2  and 
the  mean  velocity  of  the  friction  surfaces  is  given  by  the  fol- 
lowing expression: 

=  17  "  (450) 


COMBINED  CONICAL-DISC  CLUTCHES 

By  a  combined  conical-disc  clutch  is  meant  one  in  which  the 
contact  surfaces  of  the  disc  or  discs  are  conical.  Several  de- 
signs of  conical-disc  clutches  are  available,  the  most  important 
of  which  are  described  briefly  in  the  following  paragraphs. 


100 


Mean      Velocity   -  ft.   per    min. 
FIG.  240. 

307.  Hele-Shaw  Clutch. — A  sectional  view  of  the  Hele-Shaw 
multiple  conical-disc  clutch  as  used  on  motor  cars  is  shown  in 
Fig.  241.  The  driving  and  driven  discs  have  a  V-shaped  annular 
groove,  the  sides  of  which  form  the  surfaces  in  contact.  The 
phosphor-bronze  driving  discs  are  provided  with  notches  on  the 
outer  periphery  which  engage  with  suitable  projections  b  on  the 
pressed  steel  clutch  casing  a.  The  mild  steel  driven  discs  have 
notches  on  the  inner  bore  which  engage  with  the  corresponding 
projections  on  the  steel  spider  c.  This  spider  is  splined  to  the 
driving  shaft,  as  shown  in  Fig.  241.  The  V  groove  in  the  discs 
permits  a  free  circulation  of  oil,  and  at  the  same  time  insures 
fairly  rapid  dissipation  of  the  heat  generated  when  the  clutch  is 
allowed  to  slip.  The  details  of  the  mechanism  used  for  operat- 
ing the  clutch  are  shown  clearly  in  the  figure. 


440 


HELE-SHAW  CLUTCH 


[CHAP.  XVI 


Analysis  of  the  Hele-Shaw  clutch. — Since  the  surfaces  of  contact 
are  frustums  of  cones,  the  action  of  the  Hele-Shaw  clutch  is  similar 


FIG.  241. 


FIG.  242. 


to  that  of  an  ordinary  cone  clutch;  hence  the  formulas  derived  in 
Art.  295  are  applicable.  In  Fig.  242  are  shown  a  pair  of  discs  as 
used  in  the  Hele-Shaw  clutch.  Applying  the  principles  discussed 


ART.  308]  IDEAL  MULTI-CONE  CLUTCH  441 

in  Art.  295,  we  find  that  the  magnitude  of  the  moment  of  friction 
M  i  for  the  frustum  of  the  outer  cone  is 

(451) 


a 
and  that  on  the  inner  cone  is 


M,  =  (452) 


The  moment  of  friction  for  one  friction  surface  is  the  sum  of 
MI  and  Af2,  and  for  s  surfaces  the  total  torsional  moment  that 
the  clutch  is  capable  of  transmitting  is  given  by  the  following 
expression  : 


As  now  constructed,  the  number  of  discs  used  in  the  standard 
sizes  of  Hele-Shaw  motor-car  clutches  is  always  odd,  ranging 
from  15  to  33,  and  s  in  (453)  is  always  one  less  than  the  total  number 
of  discs  used. 

The  horse  power  transmitted  by  a  Hele-Shaw  clutch  may  be 
calculated  by  means  of  the  following  formula  : 


„ 

H'-  126,060  sin  a  ' 

in  which  D  denotes  the  mean  diameter  of  the  discs  as  shown  in 
Fig.  242,  and  N  denotes  the  revolution  per  minute. 

308.  Ideal  Multi-cone  Clutch.  —  A  clutch  of  the  conical-disc 
type  having  but  one  disc  was  recently  placed  on  the  market  by 
The  Akron  Gear  and  Engineering  Co.,  of  Akron,  0.  This  clutch 
is  shown  in  Fig.  243  in  the  form  of  a  friction  coupling,  connecting 
shafts  a  and  h.  The  driving  shaft  a  has  keyed  to  it  a  sleeve  b 
to  which  the  steel  casting  cone  c  is  keyed.  The  internal  surface 
of  cone  c  comes  in  contact  with  the  cone  d,  while  the  outer  sur- 
face comes  in  contact  with  the  conical  bore  of  the  casing  g.  The 
part  of  the  clutch  casing  marked  /  is  screwed  onto  the  casing  g, 
and  is  equipped  with  lugs  on  the  inner  surface.  These  lugs  cause 
the  cone  d  to  rotate  with  /,  and  at  the  same  time  permit  d  to  be 
moved  in  an  axial  direction  by  the  operating  mechanism.  To 
provide  adjustment  for  wear  at  the  contact  surfaces,  the  cone  d 
is  screwed  onto  the  ring  e.  This  ring,  held  central  by  the  casing 
/,  is  provided  with  a  series  of  slots  on  its  periphery,  into  which 
the  set  screws  I  may  be  inserted  after  the  adjustment  for  wear 
has  been  made. 


442 


MOORE  AND  WHITE  CLUTCH 


[CHAP.  XVI 


The  axial  pressure  forcing  the  cones  d,  c,  and  g  together  is  that 
due  to  a  series  of  roller  toggles  that  are  operated  by  the  sliding 
sleeve  m.  In  disengaging  the  clutch,  the  rollers  n  are  moved 
towards  the  center  of  the  shaft  and  come  in  contact  with  the 
raised  part  of  the  lugs  o,  which  are  cast  integral  with  the  ring  e. 
As  a  result,  the  cone  d  is  pulled  out  of  engagement.  Since  the 
casing  stands  idle  when  the  clutch  is  disengaged,  it  may  be  par- 
tially filled  with  oil,  thus  causing  the  driving  cone  c  to  run  in  oil 
and  insuring  good  lubrication  at  the  surfaces  in  contact. 

309.  Moore  and  White  Clutch.— In  Fig.  244  is  shown  a  friction 
coupling  in  which  the  disc  c  is  fitted  with  hardwood  blocks,  the 


FIG.   243. 


ends  of  which  are  brought  into  contact  with  the  flanged  hub  d 
and  ring  e  through  the  operation  of  the  double  toggle  mechan- 
ism. Suitable  lugs  on  the  disc  c  engage  corresponding  recesses 
on  the  flange  b,  thus  causing  c  to  rotate  with  the  latter,  and  at 
the  same  time  permitting  it  to  move  in  an  axial  direction.  The 
surface  of  the  wooden  blocks  in  contact  with  d  is  flat,  while  the 
end  in  contact  with  the  ring  e  is  in  the  form  of  a  double  cone,  as 
shown  in  the  figure.  The  clutch  is  provided  with  a  series  of 
springs  between  the  hub  d  and  ring  e,  which  prevent  excessive 
wear  of  the  friction  surfaces  when  the  clutch  is  disengaged. 
Another  form  of  combined  conical-disc  clutch,  known  as  a  slip 


ART.  309] 


MOORE  AND  WHITE  CLUTCH 


443 


gear,  is  shown  in  Fig.  152,  and  a  description  of  it  is  given  in  Art. 
231. 

The  relation  between  the  design  constant  Kz  and  the  mean 
velocity  of  the  friction  surfaces  for  the  type  of  clutch  illustrated 
in  Fig.  244  is  shown  graphically  in  Fig.  245. 


FIG.  244. 


400 


500  600  700 

Mean   Velocity  -  ft  per 

FIG.  245. 


800 
min. 


900 


1000 


RIM  CLUTCHES 

A  large  number  of  different  forms  of  rim  clutches  are  manu- 
factured, and  apparently  they  vary  only  in  the  form  of  the  rim 
or  in  the  method  of  gripping  the  rim.  A  study  of  commercial 
rim  clutches  leads  to  the  following  classification:  (a)  block; 
(b)  split-ring;  (c)  band;  (d)  roller. 


444 


EWART  CLUTCH 
BLOCK  CLUTCHES 


[CHAP.  XVI 


Block  clutches  are  used  chiefly  on  line  shafts  and  counter- 
shafts, although  there  are  several  designs  that  have  given  good 
service  on  machine  tools.  Examples  of  the  former  type  are  shown 


FIG.  246. 


in  Figs.  246,  247,  and  248,  while  the  latter  type  is  represented  in 
Fig.  249. 

310.  Transmission  Block  Clutches. — (a)  Ewart  clutch — In  Fig. 
246   are  shown   the   constructive  features   of  the   well-known 


FIG.  247. 

Ewart  clutch.  The  levers  that  move  the  friction  blocks  are 
located  inside  of  the  clutch  rim  a,  thus  decreasing  the  air  resist- 
ance at  high  speeds,  and  at  the  same  time  making  it  less  dangerous 
to  workmen  than  the  type  of  clutch  in  which  the  operating  levers 
and  links  are  exposed.  The  Ewart  clutch  is  fitted  with  either 


ART.  310] 


HUNTER  CLUTCH 


445 


two,  four,  or  six  friction  blocks,  depending  upon  the  power  that 
is  to  be  transmitted. 

(b)  Medart  clutch. — The  type  of  clutch   coupling  shown  in 
Fig.  247  differs  from  the  Ewart  clutch  in  that  the  friction  blocks 
are  of  V  shape.     Furthermore,  the  operating  levers  are  exposed, 
thus  making  this  clutch  more  or  less  dangerous.     For  trans- 
mitting large  powers,  the  Medart  clutch  is  made  as  illustrated  in 
Fig.  247,  while  for  small  powers,  the  clutch  rim  b  is  made  flat. 

(c)  Hunter  clutch.- — The   Hunter   clutch  coupling,   shown  in 
Fig.  248,  has  two  cast-iron  shoes  c  and  d  which  are  made  to  clamp 


FIG.   248. 

the  drum  b  when  the  screws  g  and  h  are  rotated  by  the  levers 
k  and  m.  Each  of  the  shoes  is  fitted  with  a  driving  pin,  by  means 
of  which  the  shoes  c  and  d  are  made  to  revolve  with  the  flanged 
hub  /  and  the  shaft  q.  The  holes  in  the  flange  /,  through  which  the 
driving  pins  pass,  are  elongated  in  order  that  the  shoes  may  move 
freely  in  a  radial  direction.  The  drum  b  is  fastened  to  the  shaft 
a  by  a  feather  key,  thus  permitting  it  to  be  drawn  out  of  contact 
with  the  shoes  c  and  d  when  the  coupling  is  not  transmitting 


446 


MACHINE-TOOL  BLOCK  CLUTCH          [CHAP.  XVI 


power.     The  levers  k  and  m  are  operated  by  the  usual  links  and 
sliding  sleeves  as  shown  in  Fig.  248. 

(d)  Machine-tool  block  clutch. — In  general,  a  block  clutch  used 
on  machine  tools  consists  of  a  shell  running  loose  on  the  shaft, 
into  which  are  fitted  two  brass  or  bronze  shoes.  These  shoes 
are  fastened  loosely  to  a  sleeve,  which  in  turn  is  splined  to 
the  shaft.  The  shoes  are  pressed  against  the  inner  surface  of 
the  shell  by  means  of  an  eccentric,  screw,  or  wedge.  Due  to 
the  compactness  of  such  clutches,  they  are  well  adapted  for  use 
where  the  space  is  limited,  as  for  example  between  the  reversing 
bevel  gears  of  a  feed  mechanism  as  shown  in  Fig.  249.  In  the 
design  illustrated  by  Fig.  249,  the  enlarged  bore  of  the  bevel  gears 
a  forms  the  shell  against  which  the  shoes  c  and  d  are  pressed  by 


FIG.  249. 

the  sliding  sleeve  /.  This  sleeve  is  integral  with  the  double 
wedges  e  that  are  fitted  to  slide  along  the  inclined  surface  of  the 
shoes  c  and  d}  as  shown  in  the  figure.  The  friction  shoes  are 
fastened  by  filister  head  machine  screws  to  the  sleeves  b  and 
therefore  rotate  with  them.  Sleeve  /  is  fastened  to  the  shaft 
by  means  of  a  feather  key.  The  shaft  g  may  serve  either  as  the 
driving  or  the  driven  member. 

311.  Analysis  of  Block  Clutches. — In  order  to  arrive  at  an 
expression  for  the  moment  of  the  frictional  resistance  of  a  block 
clutch,  some  assumption  regarding  the  distribution  of  the  con- 
tact pressure,  as  well  as  the  variation  in  the  coefficient  of  fric- 
tion, must  be  made.  As  in  the  case  of  axial  clutches,  experi- 
mental data  are  lacking,  and  in  what  follows,  we  shall  assume  that 


ART.  311] 


BLOCK  CLUTCH  ANALYSIS 


447 


the  normal  wear  at  any  point  of  the  contact  surface  is  proportional 
to  the  work  of  friction,  and  that  the  coefficient  of  friction  remains 
constant. 

(a)  Grooved  rim. — For  our  discussion,  we  shall  assume  a  block 
clutch  in  which  the  rim  is  grooved  as  shown  in  Fig.  250.  The 
total  moment  that  a  clutch  of  this  type  will  transmit  is  equal  to 
the  number  of  blocks  in  contact  multiplied  by  the  frictional 
moment  of  one  block,  the  magnitude  of  which  may  be  determined 
as  follows : 

In  Fig.  250  is  shown  a  grooved  clutch  rim  against  which  a 
single  block  is  held  by  the  force  P;  hence,  the  normal  force  acting 


FIG.  250. 


upon  an  elementary  area  of  the  surface  in  contact  is  prdddf, 
and  the  component  of  this  pressure  parallel  to  the  line  of  action 
of  the  radial  force  P  is  given  by  the  expression 


Hence 


dP  =  pr  sin/3  cos0  dd  df 
P  =  2  sin0  J  Jprcos0  d6df 


(455) 


Since  the  normal  wear  n  at  any  point  is  assumed  to  be  propor- 
tional to  the  work  of  friction,  we  get 

n  =  kpr 
If  the  surfaces  in  contact  remain  conical,  it  follows  that  the  wear 


448  BLOCK  CLUTCH  ANALYSIS  [CHAP.  XVI 

h  in  a  direction  parallel  to  the  line  of  action  of  P  is  constant; 
hence,  the  normal  wear  is 

n  =  h  sin  0  cos  0 
Combining  the  two  values  of  n  just  given,  we  obtain  the  relation 

C  cos  0 

p  =  —  -  —  '  (456) 

In  which  C  =  —  T  —     Substituting  (456)  in  (455),  and  integrating 

between  the  proper  limits,  the  following  expression  for  P  is  found  : 

P  =  2  C/sin/3  (B  +  sin  0  cos  0)  (457) 

The  moment  of  the  force  of  friction  acting  upon  the  elementary 
area  is 

dM  =  »pr2dddf  =  -^-=  rcosOdedr 
cos  |8 

The  total  moment  per  block  is  therefore: 


Combining  (457)  and  (458)  in  order  to  eliminate  the  constant  C, 
we  obtain  the  following  expression  for  the  total  moment  per 
block: 


,, 
M= 


sin0 


1 

\  (459) 


|8  L0  +  sin  0  cos  0. 

Since  (459)  gives  the  magnitude  of  the  frictional  moment  that  each 
block  will  transmit,  assuming  that  the  radial  force  per  block  is 
P,  the  total  moment  that  the  clutch  is  capable  of  transmitting  is 
obtained  by  multiplying  (459)  by  the  number  of  blocks. 

(6)  Flat  rim. — The  majority  of  the  block  clutches  in  common 
use  have  a  flat  rim;  hence  making  /?  =  90  degrees  in  (459),  the 
frictional  moment  transmitted  by  each  block  becomes 

M  =  juPD  [JS-TT;  —  | 0]  (46°) 

The  total  moment  is  obtained  in  the  same  manner  as  outlined  in 
the  preceding  paragraph. 

To  facilitate  using  (459)  and  (460),  the  function  in  the  brackets 
may  be  evaluated  for  different  angles  and  the  results  thus  ob- 
tained may  be  plotted.  Fig.  251  gives  values  of  0  4.  m0  COS0 
for  various  values  of  0. 


ART.  312]                       SPLIT-RING  CLUTCHES                                 44£ 

c 
'5 

q 

U> 

8  0.57 
.E0.56 

U) 

S>0.55 
u    0.54 

3 

n    0.53 
u 

D    OB2 

,' 

^ 

/ 

^x 

X 

x 

X 

/ 

^ 

/ 

s 

s 

/ 

X 

. 

/ 

X 

x 

x 

x 

x 

•^ 

^ 

^ 

** 

-    0.52 
> 
0.51 

^ 

-" 

' 

,-• 

^ 

^ 

20                       50                       40                       50                     "  « 

Values      o-f      6       in       Degrees 
FIG.  251. 

SPLIT-RING  CLUTCHES 

312.  Machine-tool  Split-ring  Clutches.— Split-ring  clutches 
are  used  for  all  classes  of  service  but  their  greatest  field  of  applica- 
tion appears  to  be  in  connection  with  machine  tools,  or  in  places 
where  the  diameter  of  the  clutch  as  well  as  the  space  taken  up  by 


FIG.  252. 


the  clutch  is  limited.     Such  clutches  are  shown  in  Figs.  252 
to  254,  inclusive.     An  inspection  of  these  figures  shows  that  in 


450 


SPLIT-RING  CLUTCHES  [CHAP.  XVI 


FIG.  253. 


FIG.  264. 


ART.  313]  SPLIT-RING  CLUTCH   ANALYSIS  451 

general  a  split-ring  clutch  consists  of  an  outer  shell  running  loose 
on  a  shaft  or  sleeve;  into  this  shell  is  fitted  a  split  ring.  The 
latter  may  be  expanded  by  the  action  of  a  pair  of  levers  as  shown 
in  Figs.  252  and  253,  or  by  means  of  a  wedge  as  shown  in  Fig. 
254.  A  sliding  sleeve,  operated  by  a  suitable  lever,  forms  a  con- 
venient means  of  engaging  the  split  ring  with  the  outer  shell. 
The  outer  shell  may  be  in  the  form  of  a  gear  as  shown  in  Figs. 
253  and  254,  or  it  may  form  part  of  a  pulley. 

The  well-known  Johnson  clutch  shown  in  Fig.  252  is  used  on 
countershafts  and  on  machine  tools.  It  has  been  adopted  by 
several  manufacturers  of  machine  tools  and  other  classes  of 
machinery.  The  clutch  shown  in  Fig.  253  is  that  used  by  the 
Greaves  Klusman  Tool  Co.  on  their  all-geared-head  lathe.  The 
split  clutch  represented  in  Fig.  254  is  that  used  by  the  American 
Tool  Works  on  the  double  back-gear  of  their  high-duty  lathe. 

313.  Analysis  of  a  Split-ring  Clutch.  —  (a)  Moment  of  friction.— 
For  a  split-ring  clutch,  it  seems  reasonable  to  assume  that  the 
pressure  exerted  by  the  ring  upon  the  clutch  shell  is  uniformly 
distributed  over  the  area  in  contact;  hence,  the  expression  for  the 
moment  of  the  force  of  friction  acting  upon  the  elementary  area  is 

(461) 


in  which  D  denotes  the  diameter  of  the  split  ring,  /  its  face,  and 
p  the  normal  pressure  per  unit  of  area  of  the  ring. 

The  split  ring  has  an  angle  of  contact  with  the  shell  of  some- 
what less  than  360  degrees,  but  for  all  practical  purposes  we  may 
assume  it  as  equal  to  360  degrees.  Assuming  the  coefficient  of 
friction  fj.  as  constant,  the  total  torsional  moment  transmitted 
by  the  clutch  is  obtained  by  integrating  (461).  Thus 


M  =  -  (462) 

(b)  Horse  power  transmitted.  —  The  horse  power  transmitted  by 
the  clutch  at  N  revolutions  per  minute  is 


- 


Since  ju  and  p  are  constant  for  any  given  case,  their  product  may 
be  denoted  by  a  new  constant,  as  K^.     Hence 


_ 

40,120^ 


452 


SPLIT-RING  CLUTCH  ANALYSIS          [CHAP.  XVI 


(c)  Force  required  to  spread  the  split  ring. — The  inside  diameter 
of  the  shell  of  a  split-ring  clutch  is  generally  made  %  4  to  J^2 
inch  larger  than  the  diameter  of  the  ring.  Due  to  this  fact,  a 
certain  part  PI  of  the  force  P  exerted  by  the  operating  mechanism 
is  used  in  spreading  the  ring.  As  soon  as  the  ring  comes  into  con- 
tact with  the  shell,  a  force  P2  is  required  which  will  press  the  ring 
against  the  shell,  thereby  causing  the  frictional  moment  necessary 
to  transmit  the  desired  power.  The  sum  of  PI  and  Pg  must  evi- 
dently equal  the  magnitude  of  the  force  P. 

1.  Determination  of  PI. — In  the  following  analysis  we  shall 
assume  that  the  thickness  of  the  ring  is  small  relative  to  its  radius, 
and  that  the  ring  will  readily  conform  to  the  bore  of  the  shell. 


FIG.  255. 


According  to  Bach's  "Elasticitat  und  Festigkeit,"  the  moment 
of  the  force  PI  about  the  section  at  A  in  Fig.  255  is  given  by  the 
following  expression : 

(465) 


in  which  ri  and  r2  denote  respectively  the  original  and  final  radii 
of  the  ring.     Therefore,  the  magnitude  of  PI  is 


E 


(466) 


2.  Determination  of  P2. — The  pressure  upon  an  elementary 

length  of  the  ring  is  — ~ — ,  and  the  moment  of  this  pressure  about 
z 

the  section  at  A  in  Fig.  255  is 


ART.  314] 


STUDY  OF  SPLIT-RING  CLUTCHES 


453 


dM, 


pfD*  sin  0  dB 


Integrating  between  the  proper  limits,  we  find  that  the  bending 
moment  upon  the  ring  at  the  section  through  A  has  the  following 
magnitude: 

pfD* 
2 


MA  = 


(467) 


Since  this  bending  moment  must  equal  that  due  to  the  force  P2, 
it  is  evident  that 


100 


ZOO 


300 


400 


Mean    Veloci+y    -ft.    per    min. 
FIG.  256. 


P,D  = 


from  which 


pfD 

2 


(468) 


Combining  (462)  and  (468),  the  magnitude  of  P2  in  terms  of  M, 
v,  and  D  is  as  follows  : 

M 


314.  Study  of  Split-ring  Clutches.—  From  a  study  of  a  con- 
siderable number  of  split-ring  clutches  of  different  types,  it  was 
found  that  in  nearly  all  cases  the  ring  and  shell  are  made  of  cast 
iron.  In  the  majority  of  the  designs,  the  ring  is  of  the  expanding 


454 


FA REEL  BAND  CLUTCH 


[CHAP.  XVI 


type  shown  in  Figs.  252  to  254,  inclusive.  The  contrac ting-ring 
type  is  also  used,  but  not  to  any  great  extent.  An  analysis,  simi- 
lar to  that  made  of  the  cone  and  disc  clutches,  was  made  of  a 
number  of  split-ring  transmission  clutches.  From  the  informa- 
tion furnished  by  two  manufacturers,  it  was  possible  to  determine 
the  value  of  the  design  constant  K±  for  the  various  clutches. 
The  graph  AB  of  Fig.  256  represents  the  results  obtained  on 
five  clutches  of  the  contracting  split-ring  type  made  by  one  manu- 
facturer. The  graph  CD  represents  the  results  obtained  on 
eleven  clutches  of  the  same  type  as  the  others,  but  made  by 
another  manufacturer. 


(b) 


FIG.  257. 


BAND  CLUTCHES 

Band  clutches  are  usually  installed  when  it  is  necessary  to 
transmit  heavy  loads  accompanied  by  shocks,  as  for  example,  in 
the  drives  of  rolling  mills  and  heavy  mine  hoists.  In  general,  a 
band  clutch  consists  of  a  flexible  steel  band,  either  plain  or  faced 
with  wood  or  asbestos  fabric,  one  end  of  which  is  fixed  and  the 
other  is  free  to  move  in  a  circumferential  direction.  Due  to  the 
pull  exerted  by  the  operating  mechanism  on  the  free  end  of  the 
band,  the  latter  is  made  to  grip  the  driving  or  driven  member. 

315.  Types  of  Band  Clutches. — (a)  Parrel  dutch.— A  band 
clutch  in  which  the  band  is  given  several  turns  around  the  driving 
drum  is  shown  in  Fig.  257.  In  this  design,  the  driving  drum  is 
keyed  rigidly  to  the  shaft  a  and  both  rotate  in  the  direction  indi- 
cated by  the  arrow.  The  unlined  steel  band  e  is  given  approxi- 


ART.  315]        WELLMAN-SEAVERS-MORGAN  CLUTCH 


455 


mately  six  and  one-half  turns  around  the  drum  g.  One  end  of 
this  band  is  fastened  to  the  flanged  hub  b  in  the  manner  shown 
in  Fig.  257(6),  and  the  free  end  is  operated  by  the  special  lever  d 
through  the  medium  of  the  conical  ended  shipper  sleeve  h. 

(6)  Wellman-Seavers-M organ  clutch. — Another  form  of  single 
band  clutch,  installed  on  heavy  mine  hoists  by  the  Wellman- 
Seavers-Morgan  Co.,  is  shown  in  Fig.  258.  The  band  is  lined 


FIG.  258. 

with  wood  and  has  an  angle  of  contact  on  the  clutch  ring  g  of 
approximately  300  degrees.  The  flanged  hub  6,  upon  which  the 
various  parts  of  the  clutch  proper  are  mounted,  is  keyed  to  the 
driving  shaft,  while  the  hoisting  drum,  to  which  the  clutch  ring 
is  bolted,  runs  loose  on  the  shaft. 

(c)  Litchfield  clutch. — In  Fig.  259  is  shown  a  two-band  clutch 
designed  by  the  Litchfield  Foundry  and  Machine  Co.  for  use  on 


456 


BAND  CLUTCH  ANALYSIS 


[CHAP.  XVI 


mine  hoists.     The  bands  are  lined  with  wood  and  each  band  has 
an  arc  of  contact  with  the  drum  g  approximating  140  degrees. 

316.  Analysis  of  a  Band  Clutch. — The  principles  underlying 
the  design  of  a  band  clutch  are  similar  to  those  employed  in  de- 


termining  the  power  transmitted  by  a  belt.  In  other  words,  the 
ratio  of  the  tight  to  the  loose  tensions  in  the  band  or  bands  is 
given  by  the  following  expression: 

w-**,  (470) 


ART.  317]  HORTON  ROLLER  CLUTCH  457 

in  which  T\  and  T2  denote  the  tight  and  loose  tensions,  respec- 
tively, M  the  coefficient  of  friction,  and  6  the  angle  of  contact. 
The  value  of  the  coefficient  of  friction  /x  for  a  steel  band  on  a  cast- 
iron  drum  may  be  assumed  as  0.05  when  a  lubricant  is  used,  and 
0.12  when  no  lubricant  is  used.  For  a  wood-faced  band,  fj,  may 
be  assumed  as  0.3. 

ROLLER  CLUTCH 

317.  Horton  Clutch. — The  type  of  rim  clutch  shown  in  Fig.  260 
is  known  as  the  Horton  roller  clutch,  and  is  used  to  some  extent 


FIG.  260. 

on  punching  presses.  The  cam  a  is  keyed  to  the  crankshaft,  and 
upon  its  circumference  are  cut  a  number  of  recesses  which  form 
inclined  planes.  The  rollers  d,  rolling  up  these  inclined  planes 
due  to  the  action  of  the  shell  e,  wedge  themselves  between  a  and 
the  clutch  ring  6,  thus  causing  the  crankshaft  to  rotate  with  the 
flywheel.  The  ring  b  is  keyed  to  the  flywheel  or  the  driving  gear. 
The  rollers  are  held  in  place  and  controlled  by  the  shell  e,  which 
is  connected  with  the  crankshaft  by  means  of  a  spring.  The 


458  CLUTCH  ENGAGING  MECHANISMS       [CHAP.  XVI 

latter  is  not  shown  in  the  figure.  The  lug  /  on  the  shell  e  engages 
a  latch  or  buffer  which  is  operated  by  the  treadle  on  the  machine. 
The  method  of  operation  of  this  roller  clutch  is  as  follows: 
At  the  instant  the  treadle  releases  the  shell  e,  the  spring  rotates 
the  latter  around  the  shaft  a  short  distance,  carrying  the  rollers 
with  it.  This  action  causes  the  rollers  to  wedge  between  the 
cam  a  and  the  ring  b,  thus  forming  a  rigid  connection  between 
the  flywheel  and  the  crankshaft.  To  disengage  the  clutch,  the 
treadle  is  released  and  it  in  turn  causes  the  latch  or  buffer  to 
strike  the  lug  /,  thus  forcing  the  cage  and  rollers  back  into  the 
original  position,  and  permitting  the  flywheel  to  rotate  freely 
again. 

CLUTCH  ENGAGING  MECHANISMS 

318.  Requirements  of  an  Engaging  Mechanism. — From  the 
descriptions   of    the    various    types  of    clutches   given   in  the 
preceding  articles  of  this  chapter,  it  is  evident  that  clutches 
are  engaged  by  a  lever  or  shipper  arm  through  the  medium  of  an 
engaging  mechanism  which  is  capable  of  increasing  the  leverage 
rather  rapidly  toward  the  end  of  the  lever  displacement.     In 
the  analysis  of  the  various  types  of  clutches,  the  force  required 
at  the  end  of  the  operating  lever  was  not  discussed,  since  its 
magnitude  depends  directly  upon  the  engaging  mechanism. 

In  Figs.  223,  248,  261,  and  262  are  shown  four  types  of  engag- 
ing mechanisms  in  which  the  following  important  requirements 
are  fulfilled: 

(a)  The  arc  of  lever  movement  is  not  excessive. 

(b)  The  leverage  increases  rapidly  toward  the  end  of  the  lever 
displacement. 

(c)  The  engaging  mechanism  is  self-locking,  and  therefore  no 
pawl  and  ratchet  are  necessary  to  hold  the  clutch  in  engagement. 

(d)  The  force  required  at  the  end  of  the  operating  lever  in 
order  to  engage  the  clutch  is  not  excessive.     The  magnitude  of 
this  force  may  be  assumed  to  vary  from  15  to  20  pounds  in  the 
case   of  an   overhead   clutch  installation.     For  large   clutches 
these  values  may  have  to  be  increased  somewhat. 

319.  Analysis   of  Engaging  Mechanisms. — In  the  majority 
of  engaging  mechanisms,  graphical  methods  are  generally  found 
more  convenient  for  determining  the  magnitude  of  the  force 
required  at  the  end  of  the  operating  lever.     In  Fig.  261(6)  is 


ART.  317]      ANALYSIS  OF  ENGAGING  MECHANISMS 


459 


given  the  graphical  analysis  of  the  forces  acting  on  the  mechan- 
ism shown  in  Fig.  261  (a).  The  vector  AB  represents  the  force 
Q  exerted  upon  the  spool  b  by  the  operating  lever.  Assuming 
that  the  clutch  is  provided  with  two  toggle  levers  c,  only  one 
being  shown  in  Fig.  261  (a),  the  force  exerted  by  b  upon  c 
is  represented  by  the  vector  AC.  The  lever  c  is  acted  upon  by 
three  forces  as  shown  in  the  figure.  The  magnitude  of  P  is 
represented  by  the  vector  CD.  The  analysis  of  the  forces  acting 
upon  the  second  toggle  lever  is  given  by  the  triangle  BCE,  in 
which  the  vector  EC  represents  the  magnitude  of  the  force  corre- 


sponding to  P.     The  magnitude  of  the  axial  force  produced  is 
given  by  the  vector  ED. 

Analytical  methods  sometimes  are  found  more  convenient 
than  graphical  methods,  as  the  following  analysis  will  show.  It 
is  required  to  determine  an  expression  for  the  force  P  in  terms 
of  Q  in  the  case  of  the  mechanism  shown  in  Fig.  262 (a).  This 
mechanism  is  used  on  the  split-ring  clutch  shown  in  Fig.  254. 
The  sliding  key  g  is  acted  upon  by  three  forces  as  follows:  Q, 
a  on  g,  and  /  on  g.  In  this  case,  Q  denotes  the  force  that  the 
operating  lever  exerts  upon  the  collar  j  shown  in  Fig.  254. 


460 


ANALYSIS  OF  ENGAGING  MECHANISMS       [CHAP.  XVI 


Taking  components  of  Q  and  /  on  g  in  a  direction  at  right  angles 
to  a  on  g}  we  obtain  the  relation: 

Q  cos  (p  =  (/  on  g)  sin  (a  +  2  <p) 
.  Q  cos  <p 


from  which 


(471) 


At  the  upper  end  of  the  sliding  wedge  /,  the  ring  e  produces  a 
force  that  will  act  vertically  downward  as  shown  in  Fig.  262. 
This  force  has  a  magnitude  given  by  the  expression : 

F  =  2P  tan  (8  +  <*>)  (472) 


(b) 
FIG.  262.] 

in  which  P  denotes  the  magnitude  of  the  force  tending  to  spread 
apart  the  ring  e.  Taking  components  of  F  and  /  on  g  in  a 
direction  at  right  angles  to  d  on  f,  we  have 

F  cos  <p  —  (}  on  g)  cos  (a  +  2  <p) 
from  which  we  find  that 


tan  (a 

Combining  (472)  and  (473),  we  obtain  the  following  relation 
between  P  and  Q: 


Q 


2  tan  (a  -f  2  <p)  tan  Q3  +  <p) 


(474) 


ART.  317]       ANALYSIS  OF  ENGAGING  MECHANISMS  461 

The  graphical  analysis  for  the  mechanism  illustrated  by  Fig. 
262  (a)  is  shown  in  Fig.  262(6).  The  vector  AB  represents  the 
magnitude  of  the  force  Q.  The  force  F  is  represented  by  CD, 
and  P  by  the  vector  FE. 

For  the  analysis  of  a  screw  operated  mechanism,  consult 
Art.  298. 

References 

Die  Maschinen  Elemente,  by  C.  BACH. 

Machine  Design,  Construction,  and  Drawing,  by  H.  J.  SPOONER. 

The  Gasoline  Automobile,  by  P.  M.  HELDT. 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Mechanical  Engineers  Handbook,  by  L.  S.  MARKS,  ED.  IN  CHIEF. 

Clutches  with  Special  Reference  to  Automobile  Clutches,  Trans.  A.S.M.E., 
vol.  30,  p.  39. 

Friction  Clutches  and  Their  Use,  Power,  Apr.  11  and  May  2,  1911. 

Friction  Clutch  and  Operating  Gear  for  Cruising  Engines  and  Turbines, 
Jour.  A.  S.  of  Mar.  Engr.,  vol.  26,  p.  206. 

Couplings  for  Cruising  Turbines,  London  Eng'g.,  July  4,  1913. 

Coil  Friction  Clutches,  Amer.  Mach.,  Apr.  1,  1909. 

Friction  Clutches,  Proc.  lust,  of  Mech.  Engr.,  1903. 


CHAPTER  XVII 
BRAKES 

The  function  of  a  brake  is  to  absorb  energy  by  the  creation  of 
frictional  resistance,  and  thereby  reduce  the  speed  of  a  machine 
or  bring  the  machine  to  a  state  of  rest.  The  absorbed  energy 
must  equal  that  given  up  by  the  live  load  and  all  moving  parts 
that  are  being  retarded.  Friction  in  bearings  and  between  other 
moving  parts  always  helps  a  brake. 

320.  General  Equations. — The  energy  absorbed  by  a  brake  is 
made  up  of  the  following  factors:  (1)  The  work  given  up  by  the 
live  load;  (2)  the  energy  given  up  by  the  rotating  parts.  To 
determine  an  expression  for  the  tangential  force  required  on  the 
brake  sheave  so  as  to  bring  a  load  to  rest,  we  shall  assume  the 
case  of  a  geared  hoisting  drum  lowering  a  load. 

Let  D  =  diameter  of  the  drum. 
W  =  the  load  on  the  drum. 
T  =  tangential  force  on  the  brake  sheave. 
d  =  diameter  of  the  brake  sheave. 
n  =  ratio  of  the  gearing  between  the  drum  and  the 

brake  sheave. 

t  =  number  of  seconds'  the  brake  is  applied. 
v  =  linear  velocity  of  the  load  in  feet  per  second. 
rj  =  efficiency  of  the  mechanism. 

To  bring  the  live  load  to  a  stop  in  t  seconds  requires  an  expendi- 
ture of  -=-  -  +  M  foot-pounds  of  work  at  the  drum.  In  addition 

A     L0 

to  absorbing  the  work  due  to  the  live  load,  the  brake  in  bringing 
the  machine  to  a  stop  must  also  absorb  the  kinetic  energy  of  all 
of  the  rotating  parts.  The  energy  due  to  the  rotating  parts  is 
\  Jco2,  in  which  /  is  the  moment  of  inertia  of  the  rotating  parts 
referred  to  the  axis  of  the  brake,  and  w  is  the  angular  velocity  of 
these  parts  in  radians  per  second.  It  is  usually  possible  to  obtain 
the  value  of  \  7co2  for  a  rotating  mass  having  a  complicated  form. 
The  body  may  be  divided  into  elements  in  such  a  way  that  the 

462 


ART.  320]  GENERAL  EQUATIONS  463 

kinetic  energy  of  each  element  is  easily  calculated;  then  by 
summation  the  total  kinetic  energy  is  obtained. 

Taking  into  account  the  internal  friction  of  the  machine,  the 
total  energy  to  be  absorbed  by  the  brake  in  t  seconds  is  given  by 
the  following  expression: 

Wvt   ,   Wvz      Iw2 
^  =  r7-2-+~27  +  T~  (475) 

The  work  done  by  the  tangential  force  T  in  t  seconds  is    ^  n 

foot-pounds.  Since  the  energy  given  up  by  the  load  and  rotating 
parts  must  equal  that  absorbed  by  the  brake,  we  have 

7  +  Trl 

The  minimum  value  of  the  force  T  on  the  brake  sheave  occurs 
when  the  load  has  been  brought  to  a  state  of  rest  and  its  magni- 
tude is  evidently 


321.  Classification.  —  Brakes  are  made  in  a  variety  of  forms  and 
no  definite  classification  can  be  given.     The  order  in  which  they 
are  discussed  in  this  chapter  is  as  follows: 

(a)  Block  brakes. 
(6)  Band  brakes. 

(c)  Axial  brakes. 

(d)  Mechanical  load  brakes. 

BLOCK  BRAKES 

322.  Single-  and  Double-block  Brakes.—  The  common  form 
of  block  brake  has  a  single  block  pressing  against  the  sheave,  thus 
causing  an  excessive  pressure  upon  the  shaft  bearings.     Such  a 
brake  is  shown  in  Fig.  263.     The  pressure  upon  the  shaft  bear- 
ings caused  by  the  block  in  this  form  of  brake  may  be  practically 
eliminated  by  the  use  of  two  brake  blocks  located  diametrically 
opposite  to  each  other.     An  arrangement  of  this  kind  is  used  on 
cranes,  elevators,  and  mine  hoists.     In  Figs.  264  to  267,  inclusive, 
are  shown  various  designs  of  double-block  brakes  all  of  which  are 
drawn  to  scale.     The  brakes  illustrated  by  Figs.  266  and  267 
are  used  on  mine  hoists,  and  are  commonly  called  post  brakes. 
The  first  of  these  post  brakes  was  designed  and  built  by  an  English 


464 


BLOCK  BRAKES 


[CHAP.  XVII 


manufacturer  for  a  large  mine  hoist;  while  the  second,  designed 
and  built  by  the  Nordberg  Engineering  Co.,  is  used  on  a  large 


FIG.  263. 


hoist  installed  at  the  Tamarack  mine  at  Calumet,  Mich.     As 
shown  in  Fig.  267,  the  posts  of  the  Nordberg  brake  are  held  in 


(b) 


264. 


position  by  the  swinging  links  m,  n,  and  o.  The  blocks  or  shoes 
are  made  of  steel  casting,  instead  of  wood  as  in  the  English  design 
shown  in  Fig.  266. 


465 


FIG.  266. 


466 


BLOCK  BRAKE  ANALYSIS 


[CHAP.  XVII 


323.  Analysis  of  Block  Brakes.— In  determining  the  magnitude 
of  the  forces  coming  upon  the  various  members  of  a  block  brake, 
either  algebraic  or  graphical  methods  may  be  used.  Frequently 
the  latter  save  considerable  time. 

Let  F  =  the  force  acting  at  the  fulcrum  of  the  operating 

lever. 

K  =  the  force  applied  at  the  end  of  the  operating  lever. 
P  =  the  radial  force  exerted  by  the  sheave  upon  the 

block. 

T  =  the  tangential  resistance  upon  the  block. 
At  =  the  coefficient  of  friction. 


FIG.  267. 

Since  the  action  of  the  block  upon  the  brake  sheave  is  similar 
to  the  action  between  the  shoes  and  sheave  of  a  block  clutch,  the 
various  formulas  derived  in  Art.  311  may  be  applied  directly. 
Two  cases  will  be  considered,  the  first  involving  the  action  of  a 
grooved  sheave  and  the  second  the  action  of  a  flat  sheave. 


ART.  323]  BLOCK  BRAKE  ANALYSIS  467 

(a)  Grooved  sheave.  —  A  block  brake  having  a  grooved  sheave 
is  shown  in  Fig.  263.  The  moment  of  the  frictional  resistance 
due  to  the  radial  force  P  is  given  by  the  following  expression  : 


,,       jjPd  r         sin  6         "i 

'  5^  L  e  +  sing  cos  0\  (478) 

The  product  of  the  factors  ^  and  -r-;  —  r—       —  -  may  be  treated 

v  ~\~  sin  u  cos  u 

as  a  new  factor  denoted  by  the  symbol  /*'.  This  factor  might 
be  termed  the  apparent  coefficient  of  friction  between  the  brake 
block  and  sheave. 

To  determine  an  expression  for  the  tangential  resistance  T, 
divide  the  moment  M  by  the  radius  of  the  sheave,  thus 

-  - 


To  determine  an  expression  for  the  force  K  applied  at  the  end 
of  the  operating  lever,  treat  the  latter  as  the  free  body  and  take 
moments  about  the  fulcrum.  Thus  for  the  rotation  indicated 
in  Fig.  263,  we  have 

K  =  ™^c  (480) 

To  calculate  the  size  of  the  pin  at  the  fulcrum,  we  must  de- 
termine the  magnitude  of  the  force  F  coming  upon  the  pin.  In 
general,  the  graphic  analysis  affords  the  most  direct  means  of 
determining  the  magnitude  of  F.  Treating  the  brake  lever  as 
the  free  body,  the  magnitudes  of  the  various  forces  are  readily 
obtained  by  drawing  the  force  diagram  ABDCA,  in  which  the 
vectors  BC,  CA,  AD,  and  DB  represent  the  forces  F,  K,  P,  and 
T,  respectively.  Having  determined  the  magnitude  of  the  force 
F,  the  pin  at  the  fulcrum  must  be  proportioned  so  that  it  is 
capable  of  resisting  the  bending  moment  and  bearing  pressure 
coming  upon  it. 

In  the  above  analysis,  the  frictional  resistance  on  the  brake 
shaft  was  not  considered.  The  error  in  any  case  is  small  and 
the  method  given  is  the  one  commonly  used.  However,  when 
designing  the  bearings  on  the  brake  shaft,  the  pressure  due  to 
the  force  P  should  not  be  neglected. 

(b)  Flat  sheave.  —  In  the  majority  of  installations,  the  brake 
sheave  is  made  with  a  straight  or  flat  face;  hence  in  the  preceding 
expressions  for  M  and  T  the  angle  ft  becomes  90  degrees. 
Substituting  this  value  in  (479),  we  get 

T  =  2  //P  (481) 


468  BLOCK  BRAKE  ANALYSIS  [CHAP.  XVII 

The  magnitudes  of  K  and  F  for  this  case  are  obtained  in  a  manner 
similar  to  that  given  above. 

To  facilitate  the  calculations  for  the  value  of  the  apparent 
coefficient  of  friction  //,  the  graph  given  in  Fig.  251  will  be  found 
useful. 

324.  Graphical  Analysis  of  a  Double -block  Brake. — It  is  re- 
quired to  determine  graphically  the  magnitude  of  the  force  re- 
quired to  apply  the  double-block  crane  brake  shown  in  Fig. 
264,  assuming  that  the  total  moment  of  the  frictional  resistance 
on  the  brake  sheave  is  known.  The  following  method  of 
procedure  is  suggested: 

(a)  Determine  the  relation  between  T  and  P  for  each  of  the 
blocks  by  means  of  (481);  thus 

li  =  J?  =  2M'  (482) 

fl  f% 

Having  calculated  the  value  of  //,  for  the  brake  under  discussion, 
the  actual  lines  of  action  of  the  resultant  of  T  and  P  on  each  block 
may  be  laid  off  as  shown  in  Fig.  264(6).  In  addition  to  the  re- 
sultant of  T  and  P,  each  brake  block  is  acted  upon  by  another 
force  which  is  equal  to  the  resultant  but  acts  in  the  opposite 
direction.  From  this  it  is  evident  that  each  brake  block  exerts 
a  pressure  upon  its  lever  equal  to  the  resultant. 

(6)  Each  brake  lever  is  acted  upon  by  three  forces,  as  follows : 
(1)  A  pressure  due  to  the  spring  shown  in  the  figure;  (2)  the  force 
due  to  the  brake  block;  (3)  the  reaction  at  the  fulcrum  of  the 
lever.  Of  the  three  forces  just  mentioned,  the  lines  of  action  of 
the  first  two  are  known,  also  the  magnitude  of  the  second.  Since 
the  point  of  application  of  the  third  force  is  known,  the  remaining 
unknown  properties  of  the  forces  acting  upon  the  levers  may 
readily  be  determined.  Applying  the  " triangle  of  forces"  to 
the  left-hand  lever,  the  magnitudes  of  the  various  forces  acting 
thereon  are  represented  by  the  sides  of  the  triangle  ABC  of  Fig. 
264(6).  The  vector  CA  represents  the  magnitude  of  the  spring 
pressure  KI  upon  this  lever,  as  well  as  upon  the  right-hand  lever. 
The  vector  CB  represents  the  magnitude  of  the  pressure  exerted 
upon  the  fulcrum  by  the  left-hand  lever. 

Applying  the  conditions  of  equilibrium  to  the  forces  acting  upon 
the  right-hand  lever,  we  find  that  a  couple  is  necessary  to  produce 
equilibrium.  Since  KI  =  Kz  and  RI  =  Rz,  it  is  evident  that  the 
resultant  pressure  upon  the  fulcrum,  due  to  the  right-hand  lever, 
is  equal  to  that  produced  by  the  left-hand  lever,  but  it  acts  in  the 
opposite  direction  as  shown  in  Fig.  264(6). 


ART.  325] 


SIMPLE  BAND  BRAKE 


469 


(c)  Having  determined  the  pressures  upon  the  fulcrum,  the 
dimensions  of  the  pin  may  be  calculated. 

(d)  The  operating  lever  a,  which  is  actuated  by  a  solenoid,  is 
used  to  disengage  the  brake.     Knowing  the  spring  pressure, 
the  magnitude  of  the  force  K3  is  readily  obtained  by  applying 
the  "  triangle  of  forces." 

BAND  BRAKES 

In  a  band  brake  an  iron  or  steel  band,  lined  with  wood,  leather, 
or  asbestos-fabric  encircles  the  brake  sheave  and  is  so  arranged 
that  it  may  be  tightened  or  released.  There  are  three  types  of 


FIG.  268. 

band  brakes,  as  follows:  (a)  simple;  (6)  band  brake  for  rotation 
in  both  directions;  (c)  differential. 

325.  Simple  Band  Brakes. — Two  different  designs  of  simple 
band  brakes  are  shown  in  Figs.  268  and  269.  In  general,  brakes 
of  this  type  should  be  designed  so  that  the  heaviest  pull  will 
always  come  upon  the  anchorage.  A  band  brake  arranged  in 
this  manner  requires  a  comparatively  small  pull  at  the  free  end 
of  the  band  to  apply  the  brake.  In  the  brake  shown  in  Fig.  269, 
the  band  makes  more  than  a  complete  turn  about  the  sheave  and 
for  that  reason  the  effort  required  to  apply  such  a  brake  is  small. 

Force  analysis. — The  following  method  of  procedure  for  deter- 


470 


SIMPLE  BAND  BRAKE 


[CHAP.  XVII 


mining  the  force  K  required  at  the  end  of  the  operating  lever 
is  common  to  all  of  the  simple  band  brakes. 

In  Fig.  268  assume  that  the  brake  sheave  rotates  in  the  direc- 
tion indicated  by  the  arrow  (1).  From  Art.  122,  the  ratio  of  the 
tight  to  the  loose  tension  is 

Y2  =   <  (483) 

in  which  /z  and  0  denote  the  coefficient  of  friction  and  angle  of 
contact,  respectively.     The  net  tension  on  the  brake  sheave  is 

-1)  (484) 


FIG.  269. 


Treating  the  operating  lever  as  the  free  body,  and  taking 
moments  about  the  fulcrum,  we  get 

Tb 


K 


(485) 


To  determine  the  net  cross-sectional  area  A  of  the  band,  divide 
the  maximum  tension  TI  by  the  permissible  stress  S  of  the  mate- 
rial used;  or 

A  =  (486) 


In  calculating  the  dimensions  of  the  band,  the  thickness  should 


ART  326] 


REVERSIBLE  BAND  BRAKE 


471 


not  be  made  so  great  that  a  considerable  part  of  the  force  on  the 
operating  lever  is  used  in  overcoming  the  resistance  to  bending 
of  the  band. 

The  pin  at  the  anchorage  or  fulcrum  must  be  made  of  ample 
size  to  resist  the  bending  moment  and  pressures  coming  upon  it. 

The  analysis  for  the  rotation  indicated  by  the  arrow  (2)  is 
similar  to  that  just  given,  and  is  left  to  the  student. 

326.  Band  Brakes  for  Rotation  in  Both  Directions. — As  men- 
tioned in  the  preceding  article,  the  heaviest  loaded  band  should 


FIG.  270. 

always  be  connected  to  the  anchorage.  This  condition  can  readily 
be  fulfilled  provided  the  load  acts  continuously  in  one  direc- 
tion. When  the  load  reverses  in  direction,  as  in  mine  hoists, 
cranes,  and  elevators,  the  condition  cannot  be  fulfilled.  The 
design  of  a  band  brake,  used  on  mine  hoists  and  on  the  armature 
shafts  of  motors  direct-connected  to  hoisting  drums,  is,  shown  in 
Fig.  270.  The  two  ends  of  the  band  are  connected  to  the  oper- 
ating lever  at  points  which  are  equidistant  from  the  fulcrum,  as 
shown  by  the  dimensions  6.  The  force  analysis  of  this  type  of 
brake  is  similar  to  that  given  in  Art.  325. 

327.  Differential  Band  Brakes. — The  general  arrangement  of 
a  differential  band  brake  is  shown  in  Fig.  271.     As  in  the  brake 


472 


DIFFERENTIAL  BAND  BRAKE 


[CHAP.  XVII 


illustrated  by  Fig.  270,  both  ends  of  the  band  are  connected  to 
the  operating  lever,  but  at  different  distances  from  the  fulcrum. 
(a)  Force  analysis. — Assuming  counterclock wise-rotation,  the 
magnitude  of  the  force  K  is  given  by  the  following  expression : 

Trr  hpi*0~i 

K  =  ~[^T\  (487) 

If  in  (487)  the  dimension  c  is  made  less  than  the  product  be"6,  the 
force  K  has  a  negative  value  and  the  brake  is  applied  automatic- 
ally. This  special  condition  is  used  to  a  considerable  extent  in 
connection  with  automatic  crane  brakes.  In  such  installations, 

the  automatic  band  brake  is 
used  to  take  the  place  of  the 
ordinary  ratchet  and  pawl 
mechanism.  In  Figs.  277  and 
278  are  shown  automatic  band 
brakes  fulfilling  the  function  of 
a  ratchet  and  pawl  by  permit- 
ting rotation  of  the  brake  sheave 
in  only  one  direction. 

AXIAL  BRAKES 

In  the  so-called  axial  brakes  a 
frictional  surface  of  revolution 
is  forced  against  a  correspond- 
ing surface,  the  pressure  being 
applied  in  a  direction  parallel 
to  the  axis  of  rotation.  Accord- 
ing to  the  form  of  the  surfaces 
of  revolution  in  contact,  axial  brakes  may  be  divided  into  the 
following  types:  (a)  conical  brakes;  (b)  disc  brakes. 

328.  Conical  Brakes. — One  of  the  simplest  forms  of  axial 
brake  is  the  conical  type,  the  constructive  features  of  which  are 
shown  in  Fig.  272.  The  magnitude  of  the  force  K  at  the  end  of 
the  operating  lever  may  be  determined  as  follows. 

We  shall  assume  that  the  outer  cone  forms  a  part  of,  or  is 
attached  to,  the  frame  work  of  the  machine,  while  the  inner  cone 
is  splined  to  the  rotating  shaft.  The  inner  cone  is  acted  upon  by 
the  axial  force  Q  and  the  pressure  exerted  by  the  outer  cone  upon 
the  conical  surface.  The  action  of  the  conical  brake  is  similar 


FIG.  271. 


ART.  328] 


CONICAL  BRAKES 


473 


to  the  action  of  the  cone  clutch  discussed  in  Art.  295.  According 
to  (426),  the  moment  of  friction  that  the  brake  is  capable  of  ab- 
sorbing is  given  by  the  expression 


M 


2  sin  a 
from  which  the  tangential  resistance  upon  the  cone  becomes 


(488) 


T  = 


sin  a 


(489) 


Treating  the  operating  lever  as  the  free  body  and  taking  mo- 
ments about  the  fulcrum,  we  get 


FIG.  272. 
Qb  _  Tb  sin  a 

d  Lid 


(490) 


In  some  conical  brakes  the  two  cones  are  made  of  cast  iron, 
while  in  others,  such  as  are  used  on  the  armature  shafts  of  crane 
motors,  the  outer  cone  is  of  cast  iron  and  the  inner  cone  is  faced 
with  wood.  The  angle  a  varies  from  10  to  18  degrees,  and  the 
coefficient  of  friction  may  be  assumed  as  0.12  to  0.25.  The 
former  coefficient  is  to  be  used  when  both  friction  surfaces  are 
made  of  cast  iron,  and  the  latter  when  one  of  the  surfaces  is  of 
wood  and  the  other  of  cast  iron. 

329.  Disc  Brakes. — The  disc  brake  is  simply  a  special  form  of 
the  conical  brake  having  the  cones  opened  out  into  plane  discs. 
In  practically  all  installations  of  disc  brakes,  there  are  more  than 
two  surfaces  in  contact.  A  disc  brake  having  several  contact 
surfaces  is  commonly  called  a  Weston  washer  brake.  Fig.  273 


474 


DISC  BRAKES 


[CHAP.   XVII 


shows  one  form  of  multiple-disc  brake.  The  pinion  a,  having 
a  faced  surface  at  6,  is  bushed  and  runs  loose  on  the  shaft  c.  The 
flange  d,  keyed  to  the  shaft,  has  a  faced  surface  similar  to  that 
at  b.  The  shaft  c  carries  a  ratchet  wheel  which  engages  with  a 
pawl  and  permits  rotation  in  but  one  direction.  Neither  the 
ratchet  wheel  nor  the  pawl  are  shown  in  the  figure.  Between 
the  faced  surfaces  of  the  pinion  and  the  flange  d,  a  series  of  fric- 
tion discs  is  arranged  in  such  a  way  that  alternate  discs  rotate 
with  a  and  d.  The  discs  shown  in  Fig.  273  are  of  fiber  and  steel; 
the  former  are  keyed  to  the  pinion  by  the  feather  keys  e  and 
the  latter  are  fitted  to  the  squared  hub  of  the  flange  d.  Fre- 
quently alternate  discs  of  brass  or  bronze  and  steel  are  used. 


FIG.  273. 


The  disc  brake  shown  in  Fig.  273  is  used  in  connection  with 
hoisting  machinery  when  it  is  required  to  lower  a  load  rapidly 
and  have  it  under  the  control  of  the  operator.  For  this  reason 
it  is  frequently  called  a  "dispatch  brake. "  The  shaft  being  held 
from  running  backward  by  the  ratchet  and  pawl,  the  operator 
may  lower  the  load  by  merely  unscrewing  the  handwheel  g. 
This  action  decreases  the  friction  between  the  discs  and  at  the 
same  time  releases  the  pinion.  By  screwing  up  the  handwheel 
so  as  to  increase  the  frictional  resistance  between  the  discs,  the 
speed  of  the  load  may  easily  be  controlled.  In  hoisting  the  load 


ART.  329]  DISC  BRAKES  475 

the  handwheel,  the  flange  d,  and  the  pinion  are  locked  together; 
in  other  words,  the  brake  is  converted  into  a  clutch. 

Force  analysis.  —  To  determine  the  axial  thrust  Q  and  the 
force  F  on  the  rim  of  the  handwheel  that  are  necessary  to  apply 
the  brake,  the  following  analysis  may  be  used. 

From  (442)  the  moment  of  the  frictional  resistance  of  the 
discs  is 


M  =  (491) 

in  which  s  denotes  the  number  of  friction  surfaces  and  D  the 
mean  diameter  of  these  surfaces.  The  axial  thrust  required  to 
set  the  brake  is 


To  produce  the  axial  thrust  by  means  of  the  screw  and  hand- 
wheel,  it  is  necessary  to  apply  a  force  F  on  the  rim  of  the  latter. 
The  moment  of  the  force  F  must  exceed  the  frictional  moment 
of  the  screw  plus  the  moment  of  friction  between  the  flange  d 
and  the  handwheel.  Hence 

—-  >  ^  tan  (a  +  v')  +  M',  (493) 

in  which  D'  denotes  the  diameter  of  the  handwheel;  a  the  angle 
of  thread  in  the  screw;  <p'  the  angle  of  friction  of  the  thread;  and 
M'  the  frictional  moment  between  d  and  g. 

The  moment  Ma  due  to  the  load  on  the  pinion,  must  overcome 
the  pivot  friction  between  the  pinion  and  the  thrust  washer  h, 
the  journal  friction  between  the  pinion  and  the  shaft  c,  and  the 
moment  of  friction  of  the  discs.  Denoting  the  moment  of  pivot 
friction  by  MI  and  that  of  journal  friction  by  M2,  the  magnitude 
of  Ma  is  given  by  the  following  expression: 

Ma  =  M  +  Ml  +  M2  (494) 

Substituting  the  value  of  M  from  (494)  in  (492)  ,  we  may  calculate 
the  magnitude  of  the  thrust  Q.  In  order  to  determine  the  force 
F  substitute  the  value  of  Q  in  (493). 

For  values  of  the  coefficient  of  friction  to  be  used  in  designing 
disc  brakes  those  given  in  Art.  334  (/)  are  recommended. 


476  LUDER'S  BRAKE  [CHAP.  XVII 

MECHANICAL  LOAD  BRAKES 

Mechanical  load  brakes  are  used  chiefly  in  connection  with 
chain  hoists,  winches,  and  all  types  of  crane  hoists.  In  general, 
the  functions  of  a  mechanical  load  brake  are  as  follows: 

(a)  The  brake  must  permit  the  load  to  be  raised  freely  by  the 
motor. 

(6)  It  must  be  applied  automatically  by  the  action  of  the  load 
as  soon  as  the  lifting  torque  of  the  motor  ceases  to  act  in  the 
hoisting  direction. 

(c)  It  must  permit  the  lowering  of  the  load  when  the  motor 
is  reversed.  Reversing  the  motor  releases  the  frictional  resistance 
and  allows  the  load  to  descend  by  gravity. 

Mechanical  load  brakes,  also  called  automatic  brakes,  are  made 
in  a  variety  of  forms,  but  the  greater  number  used  on  modern 
cranes  are  of  the  disc  type. 

330.  Worm-gear  Hoist  Brakes. — In  Fig.  274  are  shown  the 
constructive  features  of  two  forms  of  load  brakes  used  on  the 
worm  shaft  of  German  types  of  worm-geared  chain  hoists.  These 
brakes  are  necessary  to  prevent  the  running  down  of  the  load, 
as  the  steep  thread  angle  used  on  worms  brings  the  efficiency  of 
the  .hoist  above  60  per  cent. ;  hence  the  worm  and  its  gear  are  no 
longer  self-locking.  Brakes  similar  to  the  one  shown  in  Fig. 
274  (a)  are  also  used  on  some  American  worm-geared  types  of 
drum  hoists. 

(a)  Luder's  brake. — In  Fig.  274 (a)  is  shown  a  sectional  view 
through  Luder's  automatic  disc  brake.  The  flanged  hub  b  and 
the  cap  c  are  keyed  to  trhe  end  of  the  worm  shaft.  Between  6 
and  c,  and  rotating  upon  the  hub  of  the  latter,  is  a  hollow  bronze 
ratchet  wheel  e  which  engages  with  a  pawl.  The  latter  is  not 
shown  in  the  figure.  The  ratchet  wheel  is  made  hollow  so  as  to 
form  a  convenient  reservoir  for  the  lubricant.  Between  b  and 
e  a  leather  or  fiber  friction  disc  is  used. 

In  raising  the  load  the  friction  between  the  contact  surfaces 
of  e,  due  to  the  thrust  of  the  load  on  the  worm,  is  greater  than 
that  on  the  hard  steel  pivot  /,  and  as  a  result  the  parts  b,  e}  and 
c  rotate  with  the  shaft  a.  In  lowering  the  load  a  pawl  engages 
the  ratchet  wheel  and  holds  it  stationary,  while  the  collar  b 
and  the  cap  c  rotate  with  the  shaft,  thus  introducing  extra  fric- 
tion on  both  sides  of  e.  The  moment  of  the  frictional  resistance 
on  e  is  made  of  such  a  magnitude  as  to  prevent  the  overhauling 


ART.  330] 


BECKER'S  BRAKE 


477 


of  the  load  and  still  not  make  the  pull  on  the  hand  chain  too 
excessive. 

(b)  Becker's  brake. — The  constructive  features  of  Becker's 
automatic  conical  brake  are  shown  in  Fig.  274(6).  In  raising 
the  load  the  friction  between  the  cones  b  and  c  due  to  the  thrust 
of  the  worm  shaft  a  is  greater  than  that  between  the  screw  / 
and  the  cap  c;  hence  the  latter  rotates  with  the  shaft,  and  the 
moment  of  friction  is  reduced  to  a  minimum.  In  lowering  the 
load,  a  pawl,  not  shown  in  the  figure,  engages  the  ratchet  teeth 
e  and  prevents  the  cap  c  from  turning;  thus  the  moment  of  fric- 
tion caused  by  the  thrust  on  the  shaft  a  is  that  due  to  the  two 
cones  b  and  c.  The  moment  of  friction  of  these  cones  must  be 
made  sufficient  to  prevent  the  running-down  of  the  load,  and 
very  little  effort  is  required  to  lower  the  load  by  means  of  the 
pendant  hand  chain. 


FIG.  274. 

(c)  Force  analysis  of  Becker's  brake. — It  is  required  to  deter- 
mine an  expression  for  the  mean  diameter  D  of  the  cone  in  order 
that  the  hoist  shall  be  self-locking,  and  further  to  determine  an 
expression  for  the  moment  (P)R  that  is  required  on  the  hand 
sheave  in  order  to  lower  the  load. 

Let  Q  =  the  tangential  load  on  the  worm  gear. 
R  =  the  radius  of  the  hand  sheave. 
d  =  the  mean  diameter  of  the  worm. 
a  —  the  angle  of  the  mean  helix  of  the  worm. 
B  =  the  half  cone  angle. 
<p'  =  the  apparent  angle  of  friction  for  the  worm. 

To  prevent  the  running-down  of  the  load,  the  moment  of  the 
frictional  resistances  on  the  worm  shaft  must  exceed  the  moment 
on  the  shaft  a  due  to  the  load  Q;  hence 

*-<A  (495) 


478  NILES  BRAKE  [CHAP.  XVII 

in  which  MI  denotes  the  moment  of  friction  on  the  shaft  bearings. 
The  magnitude  of  MI  may  be  determined  provided  the  diameter 
of  the  shaft  and  the  distance  between  the  bearings  are  known. 
However,  this  moment  is  generally  small  and  for  practical  pur- 
poses may  be  neglected.  Solving  for  D  in  (495),  we  have 

N    d  tan  (a  —  p')  sin  0      2  MI  sin  0 

~r          ~w 

The  relation  for  D  given  by  (496)  must  be  fulfilled  if  the  hoist  is 
to  be  self-locking. 

The  moment  on  the  hand  chain  sheave  required  to  lower  the 
load,  assuming  the  hoist  as  self-locking,  is 


(P)R  =        rt  ~  T  tan  (a~v>}  +  M"  (497) 

in  which  Mf  denotes  the  frictional  moment  of  the  shaft  bearings. 
The  magnitude  of  Mf  may  be  determined  approximately  if  the 
diameters  of  the  shaft  bearings  are  known.  If  ball  bearings  are 
used  on  the  worm  shaft,  the  loss  due  to  the  journal  friction  will 
probably  not  exceed  3  per  cent,  of  the  total  work  expended. 
Upon  the  latter  assumption  (497)  reduces  to 

(49s) 


331.  Crane  Disc  Brakes.  —  (a)  Niles  brake.  —  In  Fig.  275  is 
shown  the  design  of  a  mechanical  load  brake  used  on  cranes 
manufactured  by  the  Niles-Bement-Pond  Co.  The  spur  gear  a 
meshes  directly  with  the  motor  pinion  and  is  keyed  to  the  sleeve 
b,  which  rotates  freely  upon  the  shaft  c.  The  one  end  of  this 
sleeve  b  is  in  the  form  of  a  two-jaw  helical  clutch  mating  with 
corresponding  helical  jaws  on  the  collar  h.  The  latter  is  keyed 
to  the  shaft,  and  to  prevent  it  from  sliding  along  the  shaft  c  an 
adjustable  thrust  collar  I  is  provided.  The  other  end  of  the  sleeve 
b  is  faced  and  bears  against  the  phosphor-bronze  disc  /.  A  simi- 
lar disc  g  is  located  between  the  ratchet  wheel  d  and  the  flange  e. 
The  latter  is  keyed  to  the  brake  shaft.  The  ratchet  wheel  d  is 
bronze  bushed  and  is  free  to  rotate  during  the  period  of  hoisting 
the  load,  but  pawls,  not  shown  in  the  figure,  prevent  rotation  of 
d  during  the  period  of  lowering  the  load.  The  pinion  p  meshes 
with  the  drum  gear. 

To  hoist  the  load,  the  motor  rotates  the  gear  a  and  the  sleeve  6 
in  the  direction  indicated  by  the  arrow,  while  the  shaft  c,  due  to 


ART.  331]        PAWLINGS  AND  HARNISCHFEGER  BRAKE          479 

the  action  of  the  load,  tends  to  turn  in  the  opposite  direction. 
The  relative  motion  between  the  helical  jaws  formed  on  6  and  h 
forces  the  sleeve  b  toward  the  flange  e,  thus  locking  the  whole 
mechanism  to  the  driving  sleeve  b.  To  lower  the  load,  the  motor 
rotates  the  sleeve  b  in  a  direction  opposite  to  that  indicated  by 
the  arrow,  thereby  reducing  the  pressure  between  the  disc  and  the 
ratchet  wheel.  Releasing  the  thrust  on  the  discs  /  and  g  permits 
the  load  to  descend  by  gravity.  As  soon  as  the  speed  of  the  shaft 
c  and  the  collar  h  exceeds  that  of  the  sleeve  6,  the  relative  motion 
between  the  helical  jaws  will  cause  an  increase  in  the  axial  thrust 
between  the  discs  and  the  ratchet  wheel,  which  in  turn  locks  the 
brake,  since  the  wheel  d  is  held  by  pawls. 


FIG.  275. 

(b)  Pawlings  and  Harnischfeger  brake. — The  constructive  fea- 
tures of  a  load  brake  used  by  the  Pawlings  and  Harnischfeger 
Co.  is  shown  in  Fig.  276.  It  differs  from  the  Niles  brake'  in  that 
the  friction  discs  /  and  g  are  made  of  fiber  instead  of  bronze,  and 
instead  of  using  a  helical  jaw  clutch  to  produce  the  thrust  upon 
the  friction  surfaces,  the  shaft  c  is  threaded  as  shown.  The 
driving  spur  gear  a,  which  meshes  with  the  motor  pinion,  has  the 
bore  of  its  hub  b  threaded  so  as  to  form  a  good  running  fit  with  the 
thread  upon  the  shaft  c.  In  general,  the  description  and  method 
of  operation  given  for  the  Niles  brake  in  the  preceding  para- 
graphs also  apply  to  the  brake  shown  in  Fig.  276. 


480 


CASE  BRAKE 


[CHAP.  XVII 


(c)  Case  brake. — Several  crane  manufacturers  are  using  load 
brakes  equipped  with  more  than  two  friction  discs.  In  Fig.  277 
is  shown  the  design  used  by  the  Case  Crane  Co.  The  spur  gear 


FIG.  276. 


a  meshes  directly  with  the  motor  pinion  and  is  keyed  to  a  flanged 
sleeve  6,  the  bore  of  which  is  threaded  so  as  to  form  a  good  work- 


Fio.  277. 


ing  fit  with  the  thread  on  the  shaft.  The  flange  of  the  sleeve  6 
bears  against  the  first  of  the  bronze  friction  discs  /.  The  flanged 
hub  e  is  keyed  to  the  shaft  c  and  bears  against  the  last  of  the 


ART.  331] 


CASE  BRAKE 


481 


bronze  discs.  The  cast-iron  friction  discs  g  are  keyed  loosely  to 
the  hub  e,  while  the  discs /are  keyed  loosely  to  the  shell  d.  The 
latter  rotates  freely  during  the  hoisting  period,  but  during  the 
lowering  of  the  load  a  properly  proportioned  differential  band 
brake  k  prevents  rotation. 

To  hoist  the  load,  the  motor  rotates  the  gear  a  and  the  sleeve 
b  in  the  direction  indicated  by  the  arrow,  while  the  shaft  c,  due 
to  the  action  of  the  load,  tends  to  turn  in  the  opposite  direction. 
Due  to  this  relative  motion,  the  threaded  sleeve  6  will  tend  to 
screw  up  on  the  shaft,  thus  clamping  the  flanges  6  and  e  to  the 
shell  d.  In  this  manner  the  whole  mechanism  is  locked  to  the 
driving  gear  a,  thus  transmitting  the  required  power  to  the  pinion 
p.  To  lower  the  load,  the  motor  rotates  the  gear  a  in  a  direction 


FIG.  278. 

opposite  to  that  indicated  by  the  arrow,  thus  tending  to  reduce 
the  axial  thrust  on  the  discs  /  and  g  and  permitting  the  load  to 
descend  by  gravity.  Should  the  speed  of  the  shaft  c,  due  to  the 
action  of  the  load,  exceed  that  of  the  gear  a,  the  resultant  relative 
motion  will  cause  the  sleeve  b  to  screw  up  on  the  shaft  and  lock 
the  brake,  since  the  reverse  rotation  of  the  shell  d  is  prevented  by 
the  differential  band  brake  k. 

The  closed  shell  d  is  made  oil  tight,  thus  assuring  lubrication 
of  the  friction  surfaces,  since  the  discs  run  in  an  oil  bath.  In 
order  to  distribute  the  oil  to  the  engaging  surfaces  all  of  the  discs 
as  well  as  the  flanges  b  and  e  are  provided  with  holes  and  grooves. 
Effective  means  of  lubricating  the  screw  threads  are  also  provided, 
as  shown  in  the  figure. 

(d)  Shaw  brake. — In  Fig.  278  is  shown  the  design  of  an  auto- 
matic multiple-disc  brake  used  by  the  Shaw  Electric  Crane  Co. 


482 


SHAW  BRAKE 


[CHAP.  XVII 


The  spur  gear  a  meshes  directly  with  the  motor  pinion  and  is 
keyed  to  the  sleeve  b,  which  rotates  freely  on  the  shaft  c.  One 
end  of  the  sleeve  b  is  in  the  form  of  a  two-jaw  helical  coupling 
mating  with  corresponding  helical  jaws  formed  on  the  flanged 
hub  e.  Cast  integral  with  the  sleeve  b  is  the  flange  h,  the  inner 
face  of  which  bears  against  the  first  of  the  cast-iron  friction  discs 
/.  The  flanged  hub  e  is  keyed  to  the  shaft  c  and  bears  against 
the  first  of  the  cast-iron  discs  g.  The  discs  /  and  g  have  lugs 
upon  their  outer  circumferences  which  fit  into  recesses  in  the 
shell  d  and  hence  must  rotate  with  d.  The  discs  ra  and  n  have 
lugs  upon  their  inner  circumferences  which  fit  into  recesses  on  the 
sleeve  b  and  hub  e,  respectively.  The  shell  d  rotates  freely  dur- 
ing the  hoisting  period,  while  during  the  lowering  of  the  load  a 
differential  band  brake  located  on  the  part  k  prevents  rotation. 


FIG.  279. 

The  operation  of  the  Shaw  brake  is  similar  to  that  given  in  detail 
for  the  Case  brake.  An  inspection  of  Fig.  278  shows  that  the 
engaging  frictional  surfaces  may  be  run  in  an  oil  bath,  and  hence 
no  trouble  should  be  experienced  as  far  as  lubrication  is  concerned. 

332.  Crane  Coil  Brakes. — A  form  of  automatic  coil  brake  using 
a  continuous  shaft  is  shown  in  Fig.  279.  This  design  has  been 
used  successfully  on  cranes  made  by  Niles-Bement-Pond  Co. 
It  consists  of  a  shell  a  carrying  at  its  closed  end  a  ratchet  wheel 
6  engaging  the  pawls  k.  One  end  of  the  bronze  coil  d  is  fixed  by 
means  of  lugs  to  the  driving  head  c,  and  the  other  end  is  fixed  to 
the  driven  head  e.  The  driving  head  c,  as  well  as  the  driving  gear 
/,  is  keyed  to  the  sleeve  g  which  rotates  freely  on  the  shaft  h.  The 
driven  head  e  is  keyed  to  the  shaft  h  and  is  provided  with  a  lug 
that  may  engage  with  a  similar  lug  on  the  sleeve  g.  These  lugs 
perform  the  function  of  establishing  a  positive  drive  between  the 


ART.  332] 


COIL  BRAKE 


483 


sleeve  g  and  the  shaft  h  in  case  the  bronze  coil  d  wears  down  too 
far  or  in  case  the  coil  breaks. 

In  hoisting  the  load,  the  gear /meshing  directly  with  the  motor 
pinion  rotates  the  head  c  as  shown  by  the  arrow  (1),  while  the 
driven  head  e  and  shaft  h  under  the  action  of  the  load  tend  to 
turn  in  the  opposite  direction,  thus  expanding  the  coil  d  against 
the  inner  surface  of  the  shell  a.  As  a  result  of  expanding  the 
coil  d,  the  whole  mechanism  is  locked  to  the  driving  head  c.  The 
motor,  in  lowering  the  load,  pulls  one  end  of  the  coil  until  the 
contact  surface  between  a  and  d  is  reduced  sufficiently  to  enable 
the  load  to  overcome  the  frictional  resistance,  thus  permitting 
the  load  to  descend  by  gravity.  It  should  be  remembered  that 
the  shell  a  is  prevented  from  rotating  in  the  reverse  direction  by 
the  ratchet  and  pawls.  The  speed  of  lowering  cannot  exceed 
that  due  to  the  motor  or  the  coil  will  expand  and  apply  the  brake. 


FIG.  280. 

333.  Cam  Brake. — The  automatic  cam  brake  shown  in  Fig.  280 
was  designed  to  replace  a  troublesome  coil  brake  of  the  two-shaft 
type.  The  shell  a  runs  free  on  both  of  the  shafts  g  and  h.  Upon 
the  closed  end  of  the  shell  is  formed  the  ratchet  wheel  b,  and  a 
suitable  pawl  prevents  rotation  of  the  shell  a  when  the  load  is 
lowered.  The  bronze  coil  originally  used  was  replaced  by  two 
brass  wings  d,  each  of  which  has  an  arc  of  contact  with  the  shell 
of  about  165  degrees.  A  spider  e,  to  which  the  wings  are  pivoted, 
is  keyed  to  the  driving  shaft  g,  and  the  cam  c  which  engages  with 
these  wings  is  keyed  to  the  driven  shaft  h. 

In  hoisting,  the  shaft  g  rotates  as  shown  by  the  arrow  (1), 
while  the  pinion  shaft  h  under  the  action  of  the  load  tends  to 
rotate  in  the  opposite  direction,  thus  causing  the  cam  c  to  force 
the  wings  d  outward  against  the  shell  and  thereby  locking  the 
complete  mechanism  to  the  driving  shaft  g.  In  lowering,  the 


484  ANALYSIS  OF  THE  SHAW  BRAKE       [CHAP.  XVII 

rotation  of  the  driving  shaft  is  reversed,  thus  tending  to  release 
the  wings  d  from  between  the  casing  a  and  the  cam  c,  and  permit- 
ting the  load  to  descend  by  the  action  of  gravity.  As  soon  as 
the  load  tends  to  run  down  too  fast,  the  cam  forces  the  wings 
outward  and  automatically  applies  the  brake. 

334.  Force  Analysis  of  an  Automatic  Brake. — In  determining 
the  relations  existing  between  the  external  forces  and  the  internal 
resistances  acting  on  an  automatic  brake,  the  following  analysis 
applied  to  the  multiple-disc  brake  shown  in  Fig.  278  may  serve 
as  a  guide. 

Let  D  =  the  mean  diameter  of  the  friction  discs. 

7  =  the  moment  of  inertia  of  the  rotating  parts  located 
between  the  load  and  the  brake,  referred  to  the  shaft 
of  the  latter. 
Q  =  the  axial  thrust  on  the  helical  jaws  during  hoisting 

period. 
(Q)  =  the  axial  thrust  on  the  helical  jaws  during  lowering 

period. 

R  =  the  pitch  radius  of  the  hoisting  drum. 
W  =  the  load  on  the  hoisting  drum. 
a  =  the  acceleration  of  the  load  while  hoisting. 
(a)  =  the  acceleration  of  the  load  while  lowering. 
d  =  the  mean  diameter  of  the  helical  jaws. 
n  =  the  gear  ratio  between  the  brake  and  the  drum. 
a  =  the  angle  of  the  helical  surface  on  the  jaws. 
(pf  =  the  angle  of  friction  for  the  helical  surfaces. 
H  =  the  coefficient  of  friction  for  the  discs. 
rj  =  the  efficiency  of  the  transmission  between  the  brake 
and  the  load. 

(a)  Axial  thrust  on  helical  jaws  for  hoisting. — During  the  hoist- 
ing period  the  action  of  the  brake  is  similar  to  that  of  a  clutch; 
hence  the  moment  M  required  on  the  gear  a  in  order  to  raise  the 
load  is 

M=|V  +  E£-|«+^  (499) 

L  g  J  nrj        K 

Equating  this  moment  to  that  of  the  internal  resistance  of  the 
discs  /  and  m,  we  get 

tan(«+V)  (500) 


ART.  334]  ANALYSIS  OF  THE  SHAW  BRAKE  485 

Combining  (499)  and  (500),  we  obtain  the  following  expression 
for  the  axial  thrust: 

2Rlw  i  ^1  i  2Ian 


_  g  R  , 

"  ' 


(6)  Condition  for  self  -locking.  —  When  the  power  is  shut  off, 
the  load  W  tends  to  run  the  brake  and  motor  in  the  reverse 
direction.  To  prevent  reversed  rotation  it  is  necessary  that  the 
moments  of  the  frictional  resistances  of  all  of  the  discs  and  the 
several  journals  shall  exceed  by  a  small  amount  the  moment 
due  to  the  load.  The  moment  of  the  load  for  the  running  down 

condition  is  -  —  ,   and  this  must'  be   somewhat  less  than  the 
n 

moment  of  friction  of  the  discs  /  and  g  and  the  shell  d,  or 

<  5  juQ'D,  (502) 

in  which  Qr  denotes  the  axial  thrust  on  the  helical  jaws.  The 
magnitude  of  Qf  may  be  determined  from  (501)  by  making  the 
acceleration  a  equal  to  zero;  hence 

,  _  2WR  ,      . 

"  ' 


To  determine  the  relation  that  must  exist  between  the  dimen- 
sions of  the  helical  jaws  and  the  friction  discs  so  as  to  satisfy  the 
condition  of  self-locking,  combine  (502)  and  (503);  whence 

d  tan  (a  +  <p')  <  ^5  (2  -  rj2)  (504) 

The  relation  expressed  by  (504)  must  be  satisfied  if  the  brake  is 
to  hold  the  load  from  running  down. 

(c)  Axial  thrust  on  helical  jaws  for  lowering.  —  If  the  power  is 
shut  off  while  the  load  is  being  lowered,  the  moment  of  the  de- 
scending load  plus  that  due  to  the  rotating  parts  tends  to  lock 
the  brake.  In  locking  the  brake,  the  external  moment  just 
mentioned  must  overcome  the  frictional  resistance  of  the  helical 
jaws  and  that  between  the  discs  g  and  n.  The  magnitude  of 

the  internal  frictional  moments  is  ~^-  (5  juD  +  d  tan  (a  +  <p')  ) 

z 

Letting  M1  denote  the  moment  due  to  the  rotating  parts  and  the 
inertia  of  the  load,  we  obtain  the  following  expression  for  the 
external  moment: 


486  ANALYSIS  OF  THE  SHAW  BRAKE       [CHAP.  XVII 

Ml  > 


from  which  («)- 


(505) 


5  juD  -h  d  tan  (a  +  ^') 
The  thrust  (Q)  becomes  a  minimum  when  the  load  comes  to 
rest  slowly,  or  in  other  words,  when  the  inertia  forces  become 
small  and  their  effect  may  be  neglected.  Making  MI  =  0  in 
(505),  the  minimum  value  of  (Q)  is  given  by  the  following 
expression  : 


d  tan  («  +  *>')) 
For  all  practical  purposes,  we  may  assume  that  (506)  gives 

the  magnitude  of  the  axial  thrust  upon  the  helical  jaws  during  the 

lowering  period. 

The  thrust  (Q)  becomes  a  maximum  when  the  motor  stops 

suddenly.     The  magnitude  of  (Q)  for  this  case  is  given  by  (505), 

in  which 

Ml  -  «?)  +  *MM  (507) 

(d)  Condition  of  self  -locking  for  lowering.  —  Assuming  that  the 
brake  is  to  be  self-locking  for  all  loads,  the  most,  unfavorable 
condition  arises  when  the  axial  thrust  is  a  minimum,  as  given  by 
(506)  .  The  resistances  that  actually  hold  the  load  from  running 
down,  assuming  the  brake  as  self-locking,  are  those  upon  the 
discs  /,  g,  m,  and  n.  Equating  the  external  moment,  due  to  the 
load  W,  to  the  Motional  moment  of  the  discs,  we  have 

(508) 


Combining  (506)  and  (508), 

d  tan  (a  +  <?')  <  5  ^D  (509) 

The  relation  given  by  (509)  must  be  fulfilled  if  the  brake  is 
to  be  self  -locking  during  the  lowering  period.  By  comparing 
(504)  and  (509),  it  follows  that  if  the  latter  is  fulfilled,  the  former 
is  also  satisfied. 

(e)  Moment  required  to  release  the  brake.  —  Again  assuming  that 
the  brake  is  self-locking,  the  motor  must  release  the  brake  in 
order  that  the  load  may  descend  by  gravity.  The  moment 
(M)  required  to  release  the  brake  must  exceed  by  a  small  amount 
the  sum  of  the  frictional  resistance  of  the  helical  jaws  and  that 
on  the  discs  /  and  m,  or 


ART.  3351  DISPOSAL  OF  HEAT  487 

M  P  +  d  tan  (a  -  *>')  (510) 


The  moment  (M)  becomes  a  maximum  when  Q'  is  maximum, 
which  occurs  directly  after  hoisting  the  load.  To  determine  this 
maximum  value  of  Q',  make  a  =  0  in  (501)  and  we  obtain  the 
relation  expressed  by  (503).  Substituting  (503)  in  (510),  the 
following  expression  for  (M)  is  obtained: 


__  . 

nrj   Ls  /*D  +  d  tan  (a  +  ?')J 

(/)  Design  constants  and  coefficients.  —  For  design  purposes, 
the  coefficient  of  friction  /*  for  various  combinations  of  materials 
may  be  assumed  as  follows: 

Wood  against  cast  iron-/*  varies  from  0.25  to  0.35. 

Cast  iron  against  cast  iron  lubricated-/*  varies  from  0.08  to  0.12. 

Cast  iron  against  bronze  lubricated-/*  varies  from  0.06  to  0.10. 

Cast  iron  against  fiber  lubricated-/*  varies  from  0.10  to  0.20. 

For  screws  and  helical  jaws  that  are  well  lubricated,  the  angle 
of  friction  <p'  may  be  assumed  as  5  degrees. 

The  angle  a  varies  from  5  to  17  degrees. 

The  axial  thrust  per  square  inch  of  projected  disc  area  varies 
between  rather  wide  limits.  An  analysis  of  twelve  brakes  of 
various  capacities  showed  that  this  pressure  varied  from  17  to 
270  pounds  per  square  inch  of  disc  area. 

The  axial  thrust  per  square  inch  of  projected  area  of  the  screw 
thread  or  helical  jaw  for  the  above-mentioned  twelve  brakes 
varied  from  90  to  1,800  pounds. 

335.  Disposal  of  Heat.  —  The  frictional  resistance  produced  by 
a  brake  generates  a  certain  amount  of  heat  which  is  equivalent 
to  the  energy  absorbed  by  the  brake.  Due  to  this  fact,  the 
brake  should  be  designed  so  that  the  heat  generated  may  be 
easily  dissipated  by  conduction  and  radiation.  Unfortunately, 
many  brakes  prove  troublesome  for  the  simple  reason  that 
the  heat  generated  is  not  dissipated  readily. 

The  rise  in  the  temperature  of  the  brake  sheave  depends  upon 
the  amount  of  energy  the  brake  is  required  to  absorb  every  time 
it  is  applied  and  upon  the  frequency  with  which  the  brake  is 
applied,  as  well  as  upon  the  weight  of  the  rim  and  the  specific 
heat  of  the  material.  In  general,  the  effect  of  the  arms  and  hub 
of  the  brake  sheave  is  neglected  in  calculating  the  rise  in 
temperature  for  a  given  case. 


488  DISPOSAL  OF  HEAT  [CHAP.  XVI! 

Prof.  Nichols,  in  his  "Laboratory  Manual  of  Physics,"  gives 
the  following  formula  for  determining  the  rise  in  temperature 
due  to  radiation: 
Let  A  =  the  area  of  the  radiating  surface  in  square  inches. 

T  =  the  number  of  minutes  the  brake  is  at  rest. 

c  =  the  mechanical  equivalent  of  the  specific  heat. 

k  =  the  radiation  factor. 

ti  =  the  lower  temperature  of  the  brake  sheave. 

tz  =  the  higher  temperature  of  the  brake  sheave. 

w  =  the  weight  of  the  brake  sheave  rim. 
Then 


log  (fc  _  ,0  =  (512) 

The  energy  absorbed  by  the  rim  is  cw  (£2  —  ti),  and  equating 
this  to  the  energy  given  up  by  the  load  and  the  rotating  parts 
as  given  by  (475),  we  obtain  the  following  expression: 

E  =  cw  fe  -  ti)  (513) 

By  means  of  (512),  the  approximate  rise  in  the  temperature  of 
the  brake  may  be  determined,  provided  the  factor  k  is  known. 
Mr.  E.  R.  Douglas,  in  an  article  entitled  "The  Theory  and  Design 
of  Mechanical  Brakes,"  published  in  the  American  Machinist 
of  Dec.  19  and  26,  1901,  states  that  "k  generally  lies  between 
0.4  and  0.8  of  a  foot-pound  of  energy  per  minute  for  each  square 
inch  of  surface  and  each  degree  Fahrenheit  which  that  surface 
is  above  the  temperature  of  the  surrounding  air."  The  lower 
temperature  t\  of  the  brake  sheave  may  be  assumed  to  vary  from 
90°  to  110°,  while  the  temperature  t2  should  not  exceed  140° 
to  200°,  depending  upon  the  material  forming  the  contact  sur- 
faces. In  order  to  prevent  charring  of  the  wood  blocks  or  leather 
and  fiber  facings,  the  temperature  tz  should  not  exceed  150°. 


References 

Die  Hebezeuge,  by  A.  ERNST. 

Die  Hebezeuge,  by  H.  BETHMANN. 

Machine  Design,  by  H.  D.  HESS. 

Magnetic  Brakes,  Amer.  Mach.,  vol.  25,  p.  523. 

Brakes  and  Brake  Mechanism,  Machinery  Reference  Series,  No.  47. 

Load  Brakes,  Amer.  Mach.,  Aug.  20,  1903. 

Principles  of  Band  Brake  Design,  Mchy.,  vol.  20,  p.  386. 

Brakes,  Mchy.t  vol.  12,  p;  619;  vol.  13,  pp.  5,  61  and  117. 


CHAPTER  XVIII 
SHAFTING 

336.  Materials. — Shafts  for  practically  all  classes  of  service  are 
subjected  to  shocks  and  jars.  During  each  revolution  the  stresses 
in  a  shaft  change  from  a  maximum  tension  to  maximum  compres- 
sion, provided  the  rotating  shaft  is  subjected  to  cross-bending. 
It  is  evident  that  the  material  for  shafting  must  be  tough  and 
ductile.  The  common  materials  used  for  shafting  are  as  follows. 

(a)  Wrought  iron. — In  the  past,  engineers  considered  a  good 
grade  of  wrought  iron  as  the  only  material  suitable  for  making 
shafts;  but  at  present,  due  to  its  excessive  cost  of  manufacture, 
wrought  iron  is  used  only  in  exceptional  cases.  Its  strength  is 
not  as  high  as  that  of  the  modern  steel  that  displaced  it. 

(6)  Bessemer  steel. — Machinery  steel  made  by  the  Bessemer 
process  is  used  quite  extensively  for  certain  classes  of  machine 
shafting.  It  is  cheap,  and  modern  methods  of  manufacture  give 
it  sufficient  ductility  and  toughness  in  the  "mild  grades"  so  that 
it  is  suitable  for  making  shafts.  One  disadvantage  of  Bessemer 
steel  is  that  it  may  contain  hidden  flaws  or  defects,  though  this 
is  not  a  very  common  occurrence;  hence  Bessemer  steel  will  ful- 
fill all  the  ordinary  requirements  in  a  large  number  of  cases. 

(c)  Open-hearth  steel. — Steel  made  by  the  open-hearth  process 
is  more  reliable  in  that  it  is  more  uniform  than  the  Bessemer  steel, 
and  for  this  reason  open-hearth  steel  is  specified  for  many  machine 
parts,  such  as  armature  shafts,  engine  shafts,  shafts  and  spindles 
of  machine  tools,  etc. 

(d)  Alloy  steels. — Many  of  the  special  steels  described  in  Chap- 
ter II  are  used  for  making  shafts  for  all  classes  of  service,  espe- 
cially when  great  strength  is  desired.     Attention  is  again  directed 
to  the  fact  that  shafts  made  from  alloy  steels  possess  no  greater 
rigidity  than  the  same  size  of  shaft  made  of  ordinary  machinery 
steel.     The  shafts  used  on  motor  cars  are  made  of  high-grade 
alloy  steels.     The  main  shafts  of  marine  and  large  hoisting 
engines  are  usually  made  of  a  high-grade  nickel  steel.     In  gen- 
eral, shafts  made  from  alloy  steels  are  more  expensive  than  those 
made  from  common  grades  of  steel. 

489 


490  COMMERCIAL  SHAFT  SIZES  [CHAP.  XV11I 

337.  Method  of  Manufacture. — Commercial  shafting  may  be 
classified  into  the  following  groups:  (a)  Turned;  (6)  cold-rolled 
or  drawn. 

(a)  Turned  shafting. — The  ingot  of  steel,  while  hot,  is  rolled 
into  bar  stock  having  a  diameter  KG  inch  greater  than  required 
for  the  finished  shaft.     The  bar  is  then  turned  down  in  a  lathe 
and  polished  accurately  to  size.     It  is  evident  that  the  diameter 
of  the  turned  shafting  is  always  KG  inch  less  than  the  so-called 
"nominal  diameter."     Large  shafts  are  forged  from  an  ingot, 
and  turned  down  and  finished  accurately  in  a  lathe. 

(b)  Cold-rolled   or   drawn   shafting.- — To    produce    cold-rolled 
shafting,  hot-rolled  bar  stock,  previously  treated  with  an  acid  so 
as  to  clean  the  outer  skin,  is  passed  through  special  rolls  under 
great  pressure,  or  drawn  through  special  dies.     This  cold-rolling 
or  drawing  process  renders  the  shaft  fairly  uniform  in  size.     The 
surface  acquires  a  polished  appearance  and  becomes  hard  and 
tough.     Experiments  on  cold-rolled  and  drawn  shafting  show 
that  the  strength  of  the  material  is  increased,  but  at  the  same  time 
the  ductility  is  reduced. 

A  disadvantage  of  cold-rolling  or  drawing  lies  in  the  fact  that 
a  considerable  amount  of  skin  tension  is  induced  in  the  material 
of  the  shaft.  This  tension  is  relieved  when  a  key  seat  is  cut, 
thus  causing  the  shaft  to  warp,  and  it  must  be  trued  up  before 
it  can  be  used.  It  is  quite  evident,  therefore,  that  neither  cold- 
rolled  nor  cold-drawn  shafting  is  well  adapted  for  use  in  high- 
grade  machinery  where  accuracy  is  desirable.  However,  for  the 
cheaper  grades  of  machinery  such  shafts  are  used  extensively. 

338.  Commercial  Sizes  of  Shafting. — Formerly,  when  wrought 
iron  was  used  for  shafting,  the  stock  sizes  of  the  hot-rolled  bars 
from  which  the  shafts  were  made  varied  by  ^-inch  increments. 
Since  these  bars  were  reduced  KG  mcn  m  finishing,  the  commer- 
cial sizes  of  shafts  thus  established  varied  by  J^-inch  increments 
but  were  always  KG  inch  less  than  each  even  %  inch  in  di- 
ameter.    Later  on,  when  steel  replaced  wrought  iron,  the  list 
of  stock  sizes  was  increased.     According  to  some  of  the  prominent 
manufacturers  of  power-transmission  machinery,  it  is  possible  to 
obtain  turned  shafting  in  the  following  sizes : 

From  %  to  2 inches,  the  diameters  vary  by  Kc-inch  increments. 

From  2  to  6  inches,  the  diameters  vary  by  ^-inch  increments. 
Sizes  which  are  KG  inch  less  than  the  even  Y±  inch  in  diameter 
are  also  obtainable. 


ART.  339]  SIMPLE  BENDING  491 

Shafts  larger  than  6  inches  in  diameter  are  usually  forged  to 
order. 

All  of  the  sizes  which  are  KG  inch  under  the  even  Y±  inch  in 
diameter  are  generally  accepted  as  standard  for  such  appurte- 
nances as  couplings,  hangers,  pillow  blocks,  etc. 

Cold-rolled  or  drawn  steel  shafting  may  be  obtained  in  sizes 
from  K6  inch  and  up,  the  diameters  varying  by  ^Q-inch 
increments. 

SHAFT  CALCULATIONS 

The  straining  actions  to  which  shafting  may  be  subjected  are 
as  follows:  (a)  simple  bending;  (6)  simple  twisting;  (c)  combined 
twisting  and  bending;  (d)  combined  twisting  and  compression. 

339.  Simple  Bending. —  In  many  classes  of  machinery,  shafts 
are  used  that  transmit  no  torsional  moment,  but  merely  support 
certain  machine  parts.  Such  shafts  may  revolve  or  remain 
stationary.  In  the  latter  case,  the  rotating  machine  parts  sup- 
ported by  the  shaft  are  generally  bronze-bushed  or  mounted  on 
ball  or  roller  bearings.  The  hoisting  drum  shown  in  Fig.  284 
is  supported  by  a  stationary  shaft  which  is  held  rigidly  by  the 
supporting  pedestals  A  and  B.  In  the  common  car  axle  we  have 
a  good  illustration  of  a  rotating  shaft  subjected  to  a  bending 
moment. 

(a)  Strength. — The  diameter  of  a  shaft  subjected  to  simple 
bending  may  be  determined  by  equating  the  external  moment  M 
to  the  moment  of  resistance  of  the  shaft.  Thus 

M  - 
from  which 

f-f- 

To  facilitate  making  calculations,  the  second  member  of  (514) 
may  be  evaluated  for  various  diameters  and  the  results  arranged 
in  chart  form  as  shown  in  Figs.  281  and  282.  The  determination 
of  the  magnitude  of  the  bending  moment  M  depends  upon  the 
number  of  bearings  supporting  the  shaft,  and  the  distribution  of 
the  forces  coming  upon  the  shaft.  The  method  of  procedure  in 
any  given  case  is  similar  to  that  used  in  the  case  of  beams.  The 
value  of  the  permissible  fiber  stress  S  varies  from  5,000  to  35,000 
pounds  per  square  inch  and  depends  upon  the  material  used  for 
making  the  shaft. 


492 


STIFFNESS  OF  SHAFTING 


[CHAP.  XVIII 


(b)  Stiffness. — In  many  machines  the  question  of  the  stiffness 
of  a  shaft  is  of  greater  importance  than  that  of  its  strength.  In 
other  words,  for  a  shaft  subjected  to  bending  only  the  transverse 
deflection  may  have  to  be  limited.  These  deflections  depend 
upon  the  method  of  supporting  the  shaft  as  well  as  the  distribu- 
tion of  the  forces  acting  on  the  shaft.  To  calculate  the  deflec- 
tions in  a  given  case  the  formulas  used  in  connection  with  beams 
will  apply.  No  definite  values  are  available  for  the  transverse 
deflections  of  machine  shafts,  as  they  depend  upon  the  service  for 


14 


1.2 


1.0 


o.a 


0.6 


0.4 


0.2 


1.4 


J.6 


1-0 


2.0 


2.2 


2.4 


2.6 


-2.6    -J 


Diameter  d    in   Inches 
FIG.  281. 

which  the  machine  is  intended.  For  line-  and  counter-shafts,  a 
transverse  deflection  of  0.01  of  an  inch  per  foot  of  length  is  con- 
sidered good  practice. 

340.  Simple  Twisting. — Shafting  is  very  rarely  subjected  to 
simple  twisting,  since  the  weights  of  pulleys  and  gears,  belt  and 
chain  pulls,  and  gear  tooth  pressures  cause  bending  stresses. 
Frequently  such  bending  stresses  are  difficult  to  determine  before- 
hand, and  due  to  the  fact  that  the  calculations  become  more  or 
less  complicated,  many  designers  omit  them  in  calculating  the 
diameter  of  shafts.  To  make  allowances  for  such  unknown  bend- 


ART.  340]  SIMPLE  TWISTING  493 

ing  moments  that  are  omitted,  a  low  fiber  stress  is  generally  used 
in  establishing  the  shaft  diameter.  Such  a  method  of  procedure 
should  seldom  be  used. 

(a)  Strength.  —  A  long  line-  or  counter-shaft  having  the  pulleys, 
gears,  or  sprockets  located  near  the  bearings  is  generally  con- 
sidered as  a  shaft  transmitting  a  simple  torsional  moment. 
Ordinarily  in  such  a  shaft,  the  belt  and  chain  pulls  are  not  exces- 
sive and  the  bending  moment  caused  by  them  may  be  omitted 
in  the  calculations  for  the  diameter  of  the  shaft.  Hence,  equat- 
ing the  torsional  moment  of  the  load  to  the  moment  of  resistance, 
we  have 


16  ' 

from  which 

I  -  w 

The  graphs  of  Figs.  281  and  282  may  be  found  convenient  in 
the  solution  of  problems  involving  the  use  of  (515),  but  it  should 

M        T 

be  remembered  that  for  the  same  diameter  of  shaft  -?r  =  S~F  • 

o       Z  o« 

The  magnitude  of  the  permissible  shearing  stress  S8  varies  from 
2,500  up. 

Substituting  in  (515)  the  value  of  T  expressed  in  terms  of  the 
horse  power  transmitted  and  the  revolutions  per  minute  of  the 
shaft,  we  obtain  the  following  expression  for  the  diameter  of  the 
shaft 


a  ==      —   -  -  ,  (516) 


in  which  H  and  N  denote  the  horse  power  and  revolutions  per 
minute,  respectively.  According  to  the  formulas  recommended 
by  several  prominent  manufacturers  of  power  transmission  ma- 
chinery, the  shearing  stress  S,  may  be  assigned  the  following 
values  : 

1  .  For  well-supported  head  shafts  carrying  main  driving  pulleys, 
sheaves,  or  gears  and  transmitting  heavy  loads,  Ss  is  approxi- 
mately 2,600. 

2.  For  regular  line  shafts  supported  on  bearings  every  8  feet, 
S8  =  4,300. 

3.  For  light  duty  line  shafts  supported  on  bearings  every  8 
or  10  feet,  S8  =  6,400. 


494 


CHART  FOR  SHAFTS 


[CHAP.  XVIII 


5          6    <V          6          9 
Diame  +  er$fd     in     Inches 
FIG.  282. 


12 


ART.  341]         COMBINED  TWISTING  AND  BENDING  495 

(b)  Stiffness.  —  In  machine  tools  it  is  necessary  that  the  im- 
portant drive  shafts  be  made  stiff  so  that  they  will  not  "  wind  up  " 
like  a  spring.  Such  angular  deflection  must  be  limited  in  machine 
tools,  while  in  other  classes  of  machinery  it  need  not  be  considered 
at  all.  To  determine  the  relation  between  the  torsional  moment 
T  and  the  angular  deflection  0,  the  following  method  may  be 
used. 

Since  the  torsional  modulus  of  elasticity  Et  represents  the 
ratio  of  the  unit  stress  to  the  unit  deformation,  we  get 


in  which  I  and  x  denote  the  length  of  the  shaft  and  the  deflection 
measured  on  the  surface  of  the  shaft,  respectively.  Both  x  and 
I  are  measured  in  inches. 

The  length  of  the  arc  x  is  ™~,  and  substituting  this  value 

in  (517),  we  obtain  the  following  expression  for  the  angular 
deflection: 

360  IS, 


Substituting  in  (518)  the  value  of  S8  obtained  from  (515),  we 
have 

584  IT 

e  =  -  (519) 


For  ordinary  shafts,  it  is  common  practice  to  limit  the  angle 
6  to  1  degree  in  a  length  of  shaft  equivalent  to  20  diameters. 

341.  Combined  Twisting  and  Bending.  —  A  rotating  shaft 
carrying  pulleys,  sprockets,  sheaves,  and  gears  is  subjected  to 
both  bending  and  twisting  when  used  for  the  transmission  of 
power.  Calculating  the  diameter  of  the  shaft  by  means  of  either 
of  the  formulas  (514)  or  (515),  and  ignoring  the  other,  would 
result  in  a  weak  shaft.  In  designing  shafts  subjected  to  com- 
bined bending  and  torsion,  several  formulas  based  upon  different 
theories  are  advocated  by  various  investigators.  The  theories 
upon  which  these  formulas  are  based  are  as  follows:  (a)  the 
maximum  normal  stress  theory;  (b)  the  maximum  strain  theory; 
(c)  the  maximum  shear  theory. 

(a)  Maximum  normal  stress  theory.  —  The  maximum  normal 
stress  or  Rankine's  theory  is  based  upon  the  assumption  that  the 


496  MAXIMUM  STRAIN  THEORY  [CHAP.  XVIII 

yield  point  depends  upon  the  maximum  normal  stress,  and  not 
upon  the  shear  or  other  stresses  acting  at  right  angles  to  it.  The 
resulting  maximum  stress  is  calculated  by  the  following  formula  : 

Max.  normal  stress  S"  =  -  +  Jsl  +  ^  (520) 

To  facilitate  the  use  of  (520)  when  designing  shafts,  it  has  been 
found  convenient  to  employ  what  is  generally  called  the  ''equiva- 
lent twisting  moment"  T",  an  expression  for  which  may  be 
derived  as  follows:  Substituting  in  (520)  the  values  of  S  and  S, 
in  terms  of  the  diameter  d,  we  obtain 


" 


S    =~(M  +  VMM1^),  (521) 

from  which 


T"  =       -P  =  M  +  \/M*  +  T2  (522) 

16 

The  so-called  equivalent  twisting  moment  T'l  will  produce  the 
same  maximum  normal  stress  as  is  produced  by  the  combined 
action  of  M  and  T.  In  using  (522),  it  is  important  to  remember 
that  S"  is  a  tensile  or  compressive  stress,  and  not  a  shearing 
stress. 

Some  designers  prefer  to  use  an  expression  for  the  "  equivalent 
bending  moment"  M"  in  place  of  (522).  Multiplying  and 
dividing  (521)  by  2,  we  obtain  the  following  expression: 


from  which 

T*)  (523) 


The  equivalent  bending  moment  Me  will  produce  the  same 
maximum  normal  stress  as  M  and  T  acting  together.  The 
allowable  stress  $"  must  be  the  same  as  that  used  with  (522). 

(b)  Maximum  strain  theory.  —  The  maximum  strain  theory, 
generally  credited  to  Saint-  Venant,  is  based  upon  the  assump- 
tion that  yielding  of  the  material  will  not  occur  until  a  certain 
deformation  has  been  produced.  To  determine  the  stress  that 
produces  yielding  according  to  this  theory,  the  following  formula 
must  be  used  : 

Max.  normal  stress  Se  =  (1  -  ra)  ^  +  (1  +  ra)  Js*  +  -£  (524) 


ART.  341]  MAXIMUM  SHEAR  THEORY  497 

in  which  the  symbol  m  denotes  Poisson's  ratio,  values  of  which 
are  given  in  Table  1.  For  steel,  m  may  be  assumed  as  0.3. 
Substituting  this  value  in  (524),  and  introducing  the  values  of 
S  and  S,  in  terms  of  d,  we  get  an  expression  for  the  equivalent 
twisting  moment  Te,  as  follows: 


Te  =  ~^==  0.70  M  +  1.3  VM2  +  T2  (525) 

The  equivalent  bending  moment  Me  becomes 


Me  =  =  0.35  M  +  0.65  \/M^+~T~2  (526) 

To  decrease  the  numerical  work  involved  in  applying  (525) 

M 
or  (526)  to  any  particular  problem,  let  -^  =  k;  then  (525)  and 

(526)  may  be  written 


Te  =    ~    =  T(Q.7k  +  1.3  AAM7!)  (527) 

and 

Me  =  7X0.35  k  +  0.65  V/c2  +  1)  (528) 

To  find  the  diameter  of  a  shaft  suitable  for  the  combined 
moments  M  and  T,  substitute  the  value  of  Te  for  T  and  Se  for 
S8  in  (515),  and  use  the  graphs  of  Figs.  281  and  282  as  directed 
in  Art.  340  (a).  If  (52&)  is  preferred,  substitute  the  value  of 
Me  for  M  and  Se  for  S  in  (514),  and  consult  the  graphs  of  Figs. 
281  and  282  for  the  diameter  of  the  shaft  corresponding  to  the 

Me 

calculated  ratio  -?r. 
o. 

(c)  Maximum  shear  theory.  —  Up  to  the  year  1900,  the  two 
theories  just  discussed  were  the  only  ones  in  use;  at  that  time  Prof. 
Guest  reported  in  the  Philosophical  Magazine  the  results  of  his 
investigations  upon  the  behavior  of  ductile  materials  subjected 
to  combined  stresses.  His  conclusion  was  that  the  yield  point 
depends  upon  the  maximum  shearing  stress;  that  is,  the  material 
yields  when  the  greatest  resultant  shear  reaches  a  certain  limit. 
The  formula  for  calculating  the  maximum  shearing  stress  that 
produces  yielding  is  as  follows: 

Max.  shear  S'e  =  ^^  &  (529) 

To  determine  the  equivalent  twisting  moment  T'e  for  this 
theory,  substitute  in  (529)  the  values  of  S  and  S8  ;  hence 


498 


SELECTION  OF  PROPER  THEORY        [CHAP.  XVIII 
ird*S' 


16 


=  VM2  +  T' 


=  T  Vk2  +  1  (530) 

To  determine  the  diameter  of  the  shaft  by  means  of  (530)  in 

any  particular  problem,  substitute   T'e    for    T   and    S'e    for    Sa 

in  (515),  and  consult  the  graphs  of  Figs.  281  and  282  as  directed 

in  Art.  340(o). 

342.  Method  of  Application. — In  order  to  determine  the 
diameter  of  a  shaft  subjected  to  combined  twisting  and  bending, 
we  must  decide  which  of  the  theories  discussed  in  Art.  341  should 
be  used.  It  should  be  noted  that  the  maximum  strain  theory 


is  really  nothing  more  than  a  refinement  of  the  maximum  normal 
stress  theory,  and  for  that  reason  is  more  accurate. 

Comparing  (527)  and  (530),  it  is  evident  that  for  equal 
diameters  d,  the  following  relation  must  exist  between  the 
allowable  stresses  and  k: 


0.7  k  +  1.3  Vk2  +  1 


s'e 


(531) 


The  relation  expressed  by  (531)  may  be  represented  graphic- 
ally as  shown  in  Fig.  283.  Every  point  on  the  curve  represents 
simultaneous  values  of  k  and  the  ratio  c  for  which  (527)  and 
(530)  will  give  the  same  shaft  diameter.  It  is  evident  that  if 
a  point  represented  by  the  coordinates  c  and  k  does  not  lie  on  the 
curve,  one  of  these  formulas  will  give  a  diameter  of  shaft  which 


ART.  343]       COMBINED  TWISTING  AND  COMPRESSION  499 

is  larger  than  that  given  by  the  other.  The  object  of  representing 
(531)  by  the  graph  of  Fig.  283  is  to  show  at  a  glance  which 
formula  or  theory  must  be  used  in  a  given  case  in  order  to  ob- 
tain the  maximum  shaft  diameter.  If  the  coordinates  c  and  k 
locate  the  point  below  the  curve,  the  maximum  strain  theory 
must  be  used;  that  is,  use  formula  (527)  or  (528).  If  the  point 
lies  above  the  curve,  the  maximum  shear  theory  or  formula 
(530)  must  be  used. 

According  to  C.  A.  M.  Smith,  an  English  investigator,  the 
ratio  of  the  working  stresses  for  mild  steel  in  tension  and  shear  is 
practically  2  to  1  instead  of  5  to  4,  as  usually  quoted  in  text  books. 
In  Table  95  are  given  the  fiber  stresses  at  the  elastic  limit  for 
tension  and  shear  as  determined  by  Prof.  Hancock. 

TABLE  95. — FIBER  STRESSES  AT  THE  ELASTIC  LIMIT 


Material 

Tension 

Shear 

Ratio  § 

Mild  carbon  steel 

47,000 

30,500 

1  54 

Nickel  steel 

76500 

38000 

2  01 

343.  Combined  Twisting  and  Compression. — Shafts  subjected 
to  a  twisting  moment  combined  with  a  compression  are  frequently 
met  with  in  machinery.  Among  the  most  common  examples  of 
such  shafts  are  those  used  for  driving  worm  gearing  and  the  pro- 
peller shafts  of  ships.  Occasionally  vertical  shafts  carrying 
heavy  rotating  parts  are  subjected  to  combined  twisting  and 
compression.  However,  in  many  cases  of  worm  gearing  the 
worm  can  be  mounted  so  that  very  little  of  the  thrust  comes 
upon  the  shaft  proper. 

(a)  Short  shaft. — The  first  case  to  be  discussed  is  one  in  which 
the  part  of  the  shaft  subjected  to  compression  is  so  short  that 
it  may  be  considered  as  a  simple  compression  member  so  far  as 
the  action  of  the  thrust  is  concerned. 

4P 

The  intensity  of  compressive  stress  for  a  solid  shaft  is  Sc  =  — -™' 

in  which  P  denotes  the  thrust.     From  (515),  the  intensity  of 
shearing  stress  due  to  the  twisting  moment  on  the  shaft  is 

16  T 
Sa  —  — -jg-  •     The  resultant  maximum  stress  due  to  the  combined 

action  of  Sc  and  S8  may  be  found  by  substituting  the  values  of 
the  latter  in  (520) ;  whence 


500       COMBINED  TWISTING  AND  COMPRESSION     [CHAP.  XVIII 

Max.  compressive  stress  =  -?-  \P  +  A/P2  +  64  T*      (532) 

Tra2  L  \  d2     J 

To  determine  the  diameter  d  of  a  shaft  having  given  the 
magnitudes  of  the  thrust  P,  the  torsional  moment  T,  and  the 
allowable  compressive  stress,  assume  a  trial  value  for  d  somewhat 
larger  than  that  required  for  the  twisting  moment  alone  and 
evaluate  (532).  If  the  calculated  value  of  the  stress  does  not 
come  near  the  allowable  maximum  make  a  second  calculation, 
and  so  on. 

(b)  Long  shaft.  —  The  second  case  to  be  considered  is  that  of  a 
shaft  in  which  the  part  subjected  to  a  thrust  is  so  long  that  it  is 
liable  to  buckle;  in  other  words  the  shaft  must  be  considered  as 
a  long  column.  According  to  Art.  15,  the  mean  intensity  of 
permissible  compressive  stress  in  a  long  column  having  a  circular 
cross-section  and  subjected  to  a  thrust  P  is  as  follows: 

o'  _     4P    _  Sc  ,_  „  . 

~  *    ~  * 


, 

"| 


Assuming  that  the  coefficient  of  elasticity  E  has  an  average 
value  of  30,000,000,  and  that  n  may  be  taken  as  unity,  (533) 
reduces  to  the  following  form  : 


i  i 


(534) 


18,500,000  d2 

The  stress  calculated  by  (534)  is  the  mean  intensity  of  com- 
pressive stress  which  corresponds  to  a  maximum  compressive 
stress  Sc  in  the  long  shaft;  hence  a  short  shaft  having  the  same 
diameter  as  the  longer  one  is  capable  of  withstanding  a  thrust 
P'  which  is  greater  than  P  in  the  ratio  of  Sc  to  S'c.  It  is  evident 
that  the  magnitude  of  the  thrust  P'  is  given  by  the  following 
expression  : 

P'=P§  (535) 

&c 

To  determine  the  diameter  of  the  shaft  necessary  to  support 
the  thrust  P  and  twisting  moment  T,  use  (532)  as  before,  but 
substitute  therein  for  P  the  magnitude  of  P'  as  calculated  by  (535). 

344.  Bending  Moments.  —  In  calculating  the  bending  moments 
coming  upon  a  shaft  supported  on  the  ordinary  type  of  bearings, 
it  seems  reasonable  to  assume  that  the  clearance  between  the 


ART.  345] 


CRANE  DRUM  SHAFT 


501 


bearings  and  the  shaft  will  permit  the  latter  to  deflect  up  to  the 
middle  of  the  bearings.  Therefore,  in  such  cases  the  moment 
arms  should  be  measured  to  the  middle  of  the  bearings,  and  a 
shaft  designed  upon  this  assumption  will  generally  be  of  ample 
size  so  far  as  strength  is  concerned. 

Whenever  a  gear,  flywheel,  or  other  machine  part  is  forced  or 
shrunk  upon  a  shaft,  it  is  practically  impossible  for  the  shaft  to 
fail  at  the  center  of  the  hub.  However,  the  shaft  may  fail  along 
a  section  near  either  end  of  the  hub,  since  any  bending  of  the 
shaft  would  tend  to  localize  the  crushing  at  those  sections.  Ac- 
cording to  Mr.  C.  L.  Griffin,  the  critical  sections  may  be  assumed 
to  lie  from  %  to  1  inch  inside  of  the  hub. 

The  majority  of  machine  designers  assume  the  moment  arms 
as  extending  to  the  center  of  hubs  and  bearings,  probably  because 


the  method  is  simple  and  the  results  obtained  are  on  the  side  of 
safety. 


SPECIAL  PROBLEMS 

The  problems  discussed  in  the  following  articles  will  serve  to 
illustrate  the  general  method  of  procedure  that  may  be  used  in 
calculating  the  bending  moments  coming  upon  a  shaft  and  finally 
in  determining  the  diameter  of  the  shaft  required  in  any  given 
case. 

345.  Crane  Drum  Shaft. — In  Fig.  284  is  shown  a  crane  drum 
running  loose  on  the  stationary  shaft.  With  this  construction 
no  twisting  moment  is  transmitted  through  the  shaft.  The  first 
step  to  be  taken  in  calculating  the  diameter  of  the  shaft  is  to 
determine  the  magnitude  and  location  of  the  maximum  bending 
moment  coming  upon  the  shaft  due  to  the  loading  shown  in  Fig. 
284, 


502  CRANE  DRUM  SHAFT  [CHAP.  XVIII 

Let   D  =  the  weight  of  the  drum. 
G  =  the  weight  of  the  gear. 
T  =  the  tooth  thrust  due  to  the  driving  pinion. 
W  =  the  load  on  each  hoisting  rope. 

Taking  moments  about  the  center  of  the  supporting  pedestal 
B,  we  have  for  the  horizontal  load  at  A  the  following  expression  : 

T 

Ah  =  j-(L  -t),  (536) 

and  for  the  vertical  load  at  A,  we  have 
A    = 


Combining  these   loads   in   the   usual   manner,    we  obtain  the 
following  expression  for  the  resultant  pressure  at  A  : 

A  =  VA*  +~Al  (538) 

Taking  moments  about  the  center  of  the  supporting  pedestal 
A,  the  horizontal  load  coming  upon  pedestal  B  is 

Tt 
B,  =  j-,  (539) 

and  the  vertical  load  at  B  is 


Hence,  the  resultant  pressure  at  B  is 


B  =  VB*h  +  El  (541) 

The  bending  moment  at  the  cejiter  of  the  bearing  C  is  Ae,  and 
that  at  the  center  of  the  bearing  D  is  Bf;  whichever  of  these 
moments  is  the  greater  must  be  used  in  calculating  the  diameter 
of  the  shaft  by  means  of  (514). 

346.  Shaft  Supporting  Two  Normal  Loads  between  the  Bear- 
ings.— A  shaft  supported  on  two  bearings  and  carrying  two  or 
more  gears,  sprockets,  or  pulleys  is  of  common  occurrence  in 
machinery.  In  some  cases  the  gears  are  located  between  the 
bearings  as  shown  in  Fig.  285,  while  in  others  they  are  arranged 
as  shown  in  Fig.  288.  Furthermore,  the  loads  coming  upon  the 
gears  or  pulleys  produce  bending  moments  that  are  either  co- 
planar  or  in  planes  inclined  to  each  other. 

(a)  Diameter  of  shaft  required  for  strength. — It  is  desired  to  de- 


ART.  346] 


SHAFT  PROBLEMS 


503 


termine  the  diameter  of  the  shaft  shown  in  Fig.  285,  assuming 
the  horse  power  H  is  transmitted  at  N  revolutions  per  minute. 
The  first  step  to  be  taken  in  the  solution  of  this  problem  is  to 
determine,  by  means  of  the  following  formula,  the  torsional 
moment  T  transmitted  by  the  shaft : 


T  =  63,030 


(542) 


Knowing  the  torsional  moment  T,  we  may  readily  calculate  the 
magnitudes  of  the  effort  P  and  the  resistance  W,  since 


T  =  PR  =  Wr 


(543) 


Having  determined  the  forces  P  and  W,  we  may  treat  the  shaft 
as  a  simple  beam  and  determine  the  bending  moments  at  impor- 
tant points  along  the  shaft.  Since  P  and  W  act  in  planes  that  are 
at  right  angles  to  each  other,  the  problem  may  be  simplified  by 


FIG.  285. 

constructing  the  bending  moment  diagram  for  each  of  these  forces 
and  later  combining  these  diagrams  in  order  to  determine  the 
maximum  moment.  In  Fig.  285  the  triangle  AEB  represents 
the  bending  moment  diagram  for  the  force  P,  and  AFB  repre- 
sents a  similar  diagram  for  the  force  W.  In  other  words,  at  the 
point  D  the  shaft  is  subjected  to  two  non-coplanar  bending  mo- 
ments; the  one  due  to  P  is  represented  by  the  vector  DE  and  the 
other  due  to  Q  is  represented  by  the  vector  DG.  These  bending 
moments  are  in  planes  at  right  angles  to  each  other;  hence  the 
resultant  moment  at  D  is  equal  to  the  vector  sum  of  DE  and  DG, 
or 


MD  = 


DG 


(544) 


504  SHAFT  PROBLEMS  [CHAP.  XVIII 

In  a  similar  manner  the  resultant  bending  moment  at  C  is  the 
vector  sum  of  CF  and  CH,  or 


Mc  =       CF  +  CH  (545) 


For  the  shaft  shown  in  the  figure,  it  is  evident  that  the  maxi- 
mum moment  occurs  under  the  pinion,  namely,  at  the  point  C. 
Having  determined  the  magnitude  of  the  maximum  bending 

M 
moment  M,  calculate  the  value  of  the  ratio  k  =  -^.     For  the 

Cf 

particular  material  used  in  the  shaft  determine  the  ratio  —7,  and 

by  means  of  the  graph  of  Fig.  283  ascertain  which  formula  must 
be  used  to  calculate  the  diameter  of  the  shaft.  The  permissible 
stress  Se  or  Se  depends  upon  the  nature  of  the  transmission  and 
the  material,  and  ordinarily  it  may  be  assumed  as  from  20  to  40 
per  cent,  of  the  stress  at  the  elastic  limit. 

(6)  Diameter  of  the  shaft  required  for  stiffness. — It  is  required  to 
determine  the  deflections  at  various  points  of  the  shaft  shown  in 
Fig.  285.  Either  the  analytical  or  the  graphical  method  may  be 
used  for  ascertaining  the  deflections,  but  since  the  loads  coming 
upon  the  shaft  are  non-coplanar  the  former  method  will  prove  to 
be  the  simpler.  From  the  theory  of  a  simple  beam  supporting  a 
load  Q,  the  deflection  AI  of  the  beam  at  any  point  Xi  inches  from 
the  left-hand  support  is  given  by  the  following  expression: 


»  -  x\  -  &*)  (546) 

The  deflection  A2  of  the  beam  at  any  point  X2  inches  from  the 
right-hand  support  may  be  calculated  by  a  formula  similar  to 
(546),  namely, 

2  -  *°  ~  a2)  (547) 


The  symbols  used  in  the  above  formulas  have  the  following 
significance:  L  denotes  the  distance  between  the  supports;  a 
the  distance  from  the  left-hand  support  to  the  load;  b  the  dis- 
tance from  the  right-hand  support  to  the  load. 

Since  the  shaft  shown  in  Fig.  285  may  be  treated  as  a  simple 
beam,  the  deflections  due  to  the  force  P  may  be  calculated  by 
means  of  (546)  and  (547).  The  deflections  due  to  the  load  W 
may  be  determined  in  the  same  manner.  Using  the  length  of  the 
shaft  as  a  base  line,  the  values  determined  by  (546)  and  (547) 


ART.  347] 


SHAFT  PROBLEMS 


505 


may  be  platted,  thus  giving  the  deflection  curve  for  each  load. 
To  determine  the  resultant  deflection  of  the  shaft  at  any  point, 
due  to  the  combined  effect  of  P  and  W,  find  the  vector  sum  of  the 
separate  deflections  corresponding  to  the  point  under  considera- 
tion. If  the  resultant  deflection  is  considered  too  great  for  the 
particular  class  of  service,  increase  the  diameter  of  the  shaft. 

347.  Shaft  Supporting  Two  Normal  Loads  with  One  Bearing 
Between  the  Loads. — (a)  Diameter  of  the  shaft  required  for  strength. 
— A  shaft,  carrying  gears  or  sprockets,  supported  on  two  bearings 
as  shown  in  Fig.  286  is  frequently  used  in  machinery.  One  of 
the  gears  is  located  outside  of  the  bearing  B,  thus  causing  a  bend- 
ing moment  at  the  center  of  this  bearing.  The  magnitude  of  the 
bending  moment  at  any  point  along  the  shaft,  due  to  the  force 
P,  may  be  obtained  by  measuring  the  ordinate  between  AD  and 


the  lines  AE  and  ED.  Thus  the  magnitude  of  the  moment  at 
B  is  represented  by  the  vector  BE.  In  a  similar  manner  the 
bending  moment  at  any  point  along  the  shaft,  due  to  the  force 
W,  may  be  determined  by  scaling  the  ordinate  between  A  B  and 
the  lines  AF  and  FB. 

Since  the  coplanar  forces  P  and  W  are  located  on  opposite  sides 
of  the  bearing  B  and  act  in  the  same  direction,  it  is  evident  that 
the  magnitude  of  the  resultant  bending  moment  at  any  point 
along  the  shaft  is  given  by  the  ordinate  between  the  lines  AFBD 
and  AED.  Combining  the  maximum  resultant  bending  moment 
with  the  torsional  moment  transmitted,  the  diameter  of  the  shaft 
may  readily  be  determined  by  the  method  outlined  in  Art.  342. 

(6)  Diameter  of  the  shaft  required  for  stiffness. — Having  deter- 
mined the  diameter  of  the  shaft  for  the  consideration  of  strength, 
it  should  be  investigated  for  stiffness.  For  the  shaft  shown  in  the 


506  SHAFT  PROBLEMS  [CHAP.  XVIII 

figure,  the  analytical  method  of  determining  the  deflections  will 
prove  simpler  than  the  graphical  method.  The  formulas  for  the 
deflection  of  the  shaft,  due  to  the  action  of  the  force  P,  are  as 
follows: 

For  any  point  on  the  shaft  at  a  distance  xi  to  the  right  of  the 
bearing  B 

At  =  ^|j  (3cxl  -  xl  +  2  cL)  (548) 

For  a  point  between  the  bearings  at  a  distance  xz  to  the  left  of 
the  bearing  B 


A,  =  -  (L  -  z2)(2L  -  z2)  (549) 


The  minus  sign  in  (549)  indicates  that  the  deflection  of  the  shaft 
between  the  bearings  is  in  the  opposite  direction  to  the  deflection 
of  that  part  of  the  shaft  overhanging  the  bearing. 

The  shaft  deflections  due  to  the  force  W  are  in  a  direction  oppo- 
site to  those  caused  by  the  force  P;  hence,  the  resultant  deflection 
at  any  point  is  the  difference  between  the  deflections  due  to  the 
loads  W  and  P.  The  deflections  between  the  bearings  due  to  W 
may  be  calculated  by  means  of  (546)  and  (547),  while  those  to 
the  right  of  the  bearing  B  are  given  by  the  following  expression  : 

Wabxl 


QEIL 


(L  +  a)  (550) 


These  deflections  may  be  represented  graphically  as  suggested 
in  the  preceding  article,  thus  showing  at  a  glance  the  location  of 
the  maximum.  If  the  maximum  deflection  exceeds  the  permissible 
value  the  shaft  diameter  must  be  increased. 

348.  Shaft  Supporting  One  Normal  and  One  Inclined  Load 
between  the  Bearings. — In  Fig.  287  is  shown  a  shaft  supported 
on  two  bearings  carrying  a  spur  and  bevel  friction  gear.  Due 
to  the  normal  pressure  Pn  between  the  contact  surfaces  of  the 
friction  gears,  the  shaft  is  subjected  to  an  axial  compression  and 
bending  moment  in  addition  to  the  bending  and  torsional  moments 
caused  by  the  tangential  forces  on  the  two  gears.  The  bending 
moments  due  to  P,  Pn,  and  W  may  be  calculated  by  the  algebraic 
method  or  they  may  be  determined  graphically.  Since  Pn  and 
W  are  coplanar  forces,  it  is  unnecessary  to  consider  each  of  them 
separately  in  determining  the  bending  moments.  The  force  and 
funicular  polygons  shown  in  Fig.  287(6)  and  (c)  are  obtained  in 


ART.  348] 


SHAFT  PROBLEMS 


507 


the  usual  manner.  Drawing  the  dividing  ray  OD  parallel  to  the 
closing  string  od,  we  obtain  in  the  vectors  CD  and  DA  the  reac- 
tions at  the  bearings  B  and  A  due  to  the  combined  action  of 
Pn  and  W.  The  force  Pn  represented  by  BF  may  be  resolved 
into  two  components  BC  and  CF  as  shown  in  the  force  polygon. 
The  component  CF  produces  a  compression  in  the  shaft  and  a 
thrust  upon  the  bearing  A.  The  magnitude  of  the  component 
BC  must  equal  the  difference  between  the  magnitudes  of  the 
pressures  BE  and  EC  produced  upon  the  shaft  at  or  near  the 


FIG.  287. 

ends  of  the  hub  of  the  bevel  friction  gear  by  the  inclined  force 
Pn.  For  all  practical  purposes  the  lines  of  action  of  the  pressures 
BE  and  EC  may  be  assumed  as  shown  in  Fig.  287  (a)  The 
magnitudes  of  BE  and  EC  may  be  determined  as  follows:  Pro- 
duce the  string  ob  until  it  intersects  the  line  of  action  be,  and  join 
this  intersection  with  that  of  the  string  oc  and  the  line  of  action 
ec;  through  the  pole  0  draw  the  ray  OE  parallel  to  the  string  oe, 
thus  establishing  the  magnitudes  of  BE  and  EC. 

To  determine  the  bending  moment  at  any  section  of  the  shaft, 
as  for  example,  along  the  line  of  action  of  the  pressure  BE, 


508 


SHAFT  PROBLEMS 


[CHAP.  XVIII 


multiply  the  ordinate  xy  of  the  funicular  polygon  by  the  pole 
distance  h.  It  should  be  remembered  that  the  ordinate  xy 
must  be  measured  to  the  scale  of  the  space  diagram,  while  the 
pole  distance  h  represents  a  force  and  hence  must  be  measured 
to  the  scale  of  the  force  diagram. 

The  tangential  force  P  causes  bending  moments  which  are  at 
right  angles  to  those  caused  by  the  force  W  and  Pn;  hence,  the 
method  given  in  Art.  346  must  be  used  to  determine  the  maxi- 
mum resultant  moment  coming  upon  the  shaft.  Instead  of 
finding  the  bending  moments  due  to  the  force  P  by  means  of  the 
graphical  method,  less  labor  is  involved  by  using  the  algebraic 
method.  If  the  shaft  is  long  relative  to  the  diameter,  it  is  neces- 
sary either  to  treat  it  as  a  long  column  or  to  change  the  location 
of  the  bearing  B .  In  other  words,  locating  the  bearing  B  ad j  acent 


to  the  back  of  the  bevel  friction  relieves  the  shaft  from  all  column 
action,  since  the  axial  component  would  be  absorbed  by  the  bear- 
ing. Such  an  arrangement  of  bearings  would  in  general  be  pre- 
ferred to  that  shown  in  Fig.  287 (a).  However,  with  the  sug- 
gested change  of  bearings  a  force  analysis  different  from  that 
given  above  would  be  necessary. 

349.  Two -bearing  Shaft  Supporting  Three  Loads. — Frequently 
shafts  supported  on  two  bearings  and  carrying  more  than  two 
gears,  sheaves,  or  sprockets  are  required.  Such  a  shaft  support- 
ing three  loads  is  shown  in  Fig.  288.  The  bending  moment 
diagram  due  to  the  load  P  on  the  large  driving  gear  is  represented 
by  the  triangle  AEB,  while  that  due  to  the  loads  W  acting  on 
the  overhanging  pinions  is  represented  by  CFGD.  Since  P 
and  W  are  non-coplanar,  the  method  of  procedure  for  determin- 
ing the  required  diameter  of  the  shaft  is  similar  to  that  given  in 


ART.  350]  HOLLOW  SHAFTS  509 

Art.  346.  Having  calculated  the  shaft  diameter  necessary  for 
strength,  the  deflection  must  be  investigated.  For  the  shaft 
under  consideration  the  deflections  at  several  points  along  the 
shaft  are  readily  obtained  by  the  algebraic  method,  after  which 
the  deflection  curves  for  the  two  systems  of  loads  may  be  plotted. 
The  maximum  resultant  deflection  at  any  point  may  be  de- 
termined from  the  curves,  and  if  this  is  found  excessive  the 
diameter  of  the  shaft  must  be  increased. 

The  deflection  at  any  point  between  the  overhanging  load  and 
the  adjacent  bearing,  due  to  the  action  of  the  two  equal  loads 
W,  may  be  calculated  by  the  expression 

AY  =  ~  (3  a(L  +  x,)  -  *?),  (551) 

in  which  x\  denotes  the  distance  from  the  bearing  to  the  point 
under  discussion.  For  the  part  between  the  bearings,  the  de- 
flection at  any  point  due  to  the  loads  W  is 

-  *,),  (552) 


in  which  x2  denotes  the  distance  from  the  bearing  to  the  point 
considered. 

350.  Hollow  Shafts.  —  In  any  shaft  the  outer  fibers  of  the 
material  are  more  useful  in  resisting  the  bending  and  twisting 
than  the  fibers  at  or  near  the  center;  hence  the  material  may 
be  distributed  more  efficiently  by  making  the  shafts  hollow. 
Furthermore,  the  weight  of  such  a  shaft  is  diminished  in  greater 
proportion  than  its  strength. 

It  is  evident  that  the  strength  of  a  hollow  shaft  is  equivalent 
to  the  strength  of  the  solid  shaft  minus  the  strength  of  the 
shaft  having  a  diameter  equal  to  the  diameter  of  the  hole.  In 
determining  the  strength  of  the  shaft  having  a  diameter  equal 
to  that  of  the  hole,  the  fiber  stress  to  be  used  must  be  that 
produced  in  the  solid  shaft  at  a  point  whose  distance  from  the 
center  is  equal  to  the  radius  of  the  hole.  Letting  S8  denote  the 
shearing  stress  produced  in  the  outer  fiber  of  the  shaft  having  a 

diameter  dlt  the  stress  produced  at  a  distance  -~  from  the  center 
of  the  shaft  is  ^  S.. 


510  HOLLOW  SHAFTS  [CHAP.  XVIII 

(a)  Torsional  strength.  —  For  a  hollow  shaft  the  relation  between 
the  twisting  moment  and  the  diameters  of  the  shaft  is 


(553) 


Denoting  the  ratio  of  dz  to  di  by  u,  the  expression  for  the  large 
diameter  of  a  hollow  shaft  becomes 


T  ,«.,. 

(554) 


(6)  Torsional  stiffness.  —  The  angular  deflection,  in  degrees, 
caused  by  a  given  torsional  moment  T  may  be  calculated  by 
means  of  the  following  formula,  obtained  from  (518)  and  (553) 
by  eliminating  S8: 

584  IT 
=  Et(d\  -  di} 

(c)  Transverse  strength.  —  Occasionally  it  may  be  desired  to 
calculate  the  diameter  of  a  hollow  shaft  subjected  to  a  given  bend- 
ing moment  M.  This  may  be  done  by  the  use  of  the  following 
formula,  which  is  obtained  by  equating  M  to  the  moment  of 
resistance  of  the  hollow  shaft  and  solving  for  d±: 


10.2  M 


(d)  Hollow  and  solid  shafts  of  equal  strength.  —  It  is  required  to 
determine  the  relation  between  the  diameter  of  a  solid  shaft  and 
that  of  a  hollow  shaft  of  the  same  strength.  Equating  the  mo- 
ments of  resistance  to  twisting  for  the  two  shafts,  we  obtain 


or 

(557) 


For  the  same  strength,  a  hollow  shaft  is  much  lighter  than  a 
solid  one.  The  per  cent,  saved  in  weight  is  given  by  the  following 
formula  : 

per  cent,  gain  =   T—  "  ^  +  d'j  100  (558) 


ART.  351]  EFFECT  OF  KEY-SEATS  511 

EFFECT  OF  KEY-SEATS  ON  SHAFTING 

The  effect  of  a  key-seat  in  a  shaft  is  to  decrease  slightly  both  its 
strength  and  stiffness.  In  order  to  obtain  some  knowledge  as  to 
the  extent  of  this  change  in  strength  and  stiffness,  Prof.  H.  F. 
Moore  of  the  University  of  Illinois  made  a  series  of  tests,  the 
results  of  which  were  reported  in  Bulletin  No.  42,  University  of 
Illinois  Experiment  Station.  The  shafts  used  in  these  tests 
varied  in  diameter  from  1%  to  2J4  inches  inclusive.  Both  cold- 
rolled  and  turned  shafts  made  of  soft  steel  were  tested.  The 
key-seats  cut  into  these  shafts  were  of  common  proportions. 

351.  Effect  upon  Strength. — According  to  the  results  obtained 
by  Prof.  Moore,  the  ultimate  strength  of  a  key-seated  shaft  is 
practically  the  same  as  the  ultimate  strength  of  the  solid  shaft. 
Furthermore,  very  little  difference  was  observed  between  the 
strength  of  shafts  with  short  key-seats  and  of  similar  shafts  hav- 
ing long  key-seats.     The  tests,  however,  showed  conclusively 
that  a  key-seat  has  a  decided  influence  upon  the  elastic  strength 
of  the  shaft.     In  order  to  put  the  results  of  these  experiments 
into  usable  torn,  the  so-called  "efficiency  of  the  shaft"  was  deter- 
mined for  each  size  of  shaft  tested.     By  the  term  " efficiency"  is 
meant  the  ratio  of  the  elastic  strength  of  the  shaft  with  the  key- 
seat  to  the  elastic  strength  of  the  solid  shaft.     The  following 
equation  for  the  efficiency  is  suggested  by  Prof.  Moore  as  repre- 
senting fairly  well  the  results  he  obtained: 

Ei  =  1.0  -  0.2  w  -  1.1  h,  (559) 

in  which  w  denotes  the  ratio  of  the  width  of  the  key-seat  to  the 
shaft  diameter,  and  h,  the  ratio  of  the  depth  of  the  key-seat  to 
the  diameter  of  the  shaft. 

352.  Effect  upon  Stiffness. — A  number  of  tests  were  also  made 
to  determine  the  effect  of  key-seats  upon  the  angular  stiffness 
of  shafts.     The  following  equation  for  the  ratio  of  the  angle  of 
twist  of  the  key-seated  shaft  to  the  angle  of  twist  of  the  solid 
shaft  is  due  to  Prof.  Moore,  and  may  serve  as  a  guide  in  determin- 
ing the  probable  weakening  effect  the  key-seat  has  upon  the  tor- 
sional  stiffness  of  the  shaft. 

E2  =  1.0  +  0.4  w  +  0.7  h  (560) 


512  REFERENCE  [CHAP.  XVIII 

References 

Manufacture  of  Cold  Drawn  Shafting,  Amer.  Mach.,  vol.  41,  p.  89. 

Production  of  Small  Hollow  Shafting,  Amer.  Mach.,  vol.  41,  p.  367. 

Machinery  Shafting,  Machinery  Reference  Series,  No.  12. 

Heavy  Duty  Shafts  with  Two  and  Three  Bearings,  Mchy.,  vol.  20,  p.  659. 

Torque  of  Propeller  Shafting,  London  Eng'g.,  Apr.  12,  1907. 

Stresses  and  Deflections  of  Shafts,  Amer.  Mach.}  vol.  37,  p.  1027,  and  vol. 
38,  p.  10. 

Charts  for  Critical  Speeds,  Amer.  Mach.,  vol.  40.  p.  809. 

Critical  Speeds  of  Shafts,  Amer.  Mach.,  vol.  45,  p.  505. 

Critical  Speeds  of  Rotors  Resting  on  Two  Bearings,  Amer.  Mach.,  vol. 
46,  p.  97. 

Critical  Speeds  of  Rotors  Resting  on  Three  Bearings,  Amer.  Mach.,  vol.  46, 
p.  193. 

Critical  Speed  Calculations,  Jour.  A.  S.  M.  E.,  June,  1910. 

Centrifugal  Whirling  of  Shafts,  Trans.  Royal  Soc.,  vol.  185  A,  pp.  279- 
360. 


CHAPTER  XIX 

BEARINGS  AND  JOURNALS 

BEARINGS 

353.  Types  of  Bearings. — Bearings  may  be  divided  into  two 
general  classes:,  (a)  sliding;  (6)  rolling. 

(a)  Sliding  bearings. — The  sliding  bearings  in  common  use  in 
machinery  are  of  three  types.  The  first  type,  called  right  line 
bearing,  is  one  in  which  the  motion  is  parallel  to  the  elements  of 
the  sliding  surfaces.  These  sliding  surfaces  may  be  flat,  as  the 
guides  on  engine  crossheads  and  the  ways  of  large  planers  and 
milling  machines,  or  they  may  be  angular  as  the  ways  on  small 
planers  and  lathes.  Circular  guides  are  also  in  use  for  the  cross- 
heads  of  engines  and  spindles  of  boring  and  drilling  machines. 

The  second  type  of  bearing,  called  a  journal  bearing  consists  of 
two  machine  parts  that  rotate  relatively  to  each  other.  The 
part  which  is  enclosed  by  and  rubs  against  the  other  is  called 
the  journal,  and  the  part  which  encloses  the  journal  is  called 
the  box  or  less  specifically  the  bearing.  In  the  more  common 
form  of  journal  bearings,  the  journal  rotates  inside  of  a  fixed 
bearing.  In  some  cases,  as  in  a  loose  pulley  or  a  hoisting  drum, 
the  journal  is  fixed  and  the  bearing  rotates,  while  in  other  cases 
both  the  journal  and  the  bearing  have  a  definite  motion,  as  for 
example,  a  crankpin  and  its  bearing  in  the  connecting  rod. 

The  thrust  bearing  is  the  third  type  of  sliding  bearing.  It  is 
used  to  take  the  end  thrust  in  bevel  and  worm  gearing,  or  in 
general  any  force  acting  in  the  direction  of  the  shaft  axis.  Thrust 
bearings  are  of  two  kinds:  (1)  The  so-called  step-  or  pivot-bearing, 
which  supports  the  weight  of  a  vertical  shaft  and  its  attached 
parts.  The  shaft  terminates  in  the  bearing.  (2)  The  collar 
thrust  bearing,  which  is  used  on  propeller  shafts,  spindles  of  drill 
presses,  and  shafts  carrying  bevel  and  worm  gears.  In  such 
cases  the  shaft  generally  extends  through  and  beyond  the  bearing. 

(6)  Rolling  bearings. — Rolling  bearings  include  all  bearings  in 
which  rolling  elements  are  used  for  supporting  the  rotating  mem- 
bers. This  class  of  bearings  is  discussed  in  detail  in  Chapter  XX. 

513 


514  BEARING  MATERIALS  [CHAP.  XIX 

JOURNAL  BEARING  CONSTRUCTION 

354.  General  Considerations. — In  designing  bearings  the  fol- 
lowing important  points  must  be  given  due  consideration. 

(a)  The  proper  bearing  material  must  be  selected  with  respect 
to  the  load  coming  upon  the  bearing  and  the  material  used  for 
the  journal. 

(6)  Provision  must  be  made  for  anchoring  the  bearing  material 
to  the  bearing  proper. 

(c)  Provision  must  be  made  for  taking  up  any  wear  that  is 
liable  to  occur. 

(d)  In  many  cases  means  must  be  provided  for  preserving  the 
alignment  of  the  bearings. 

(e)  Proper  clearance  between  the  journal  and  its  bearing  must 
be  provided. 

(/)  Means  must  be  provided  for  lubricating  the  bearing. 

(g)  Bearings  running  at  high  speeds  and  subjected  to  high 
pressures  must  be  provided  with  some  means  of  dissipating  the 
heat  that  is  generated  by  friction. 

(ti)  The  dimensions  of  the  bearing,  that  is,  the  diameter  and 
the  length,  are  fixed  by  the  journal  with  which  the  bearing  is  to 
run.  The  proportions  of  the  journal  are  determined  from  a  con- 
sideration of  strength,  rigidity,  rubbing  speed,  and  the  permissi- 
ble pressure  per  square  inch  of  projected  area. 

355.  Selection  of  Bearing  Materials. — As  a  rule  bearings  give 
the  best  service  when  the  material  of  the  bearing  and  that  of  the 
journal  are  unlike.     No  satisfactory  explanation  has  ever  been 
offered  why  unlike  materials  are  better,  but  it  is  claimed  that 
with  like  materials  the  frictional  resistance  and  the  wear  are 
greater.     However,  there  are  exceptions,  as  under  certain  condi- 
tions hardened  steel  against  hardened  steel,  and  cast  iron  in  con- 
tact with  cast  iron  have  given  excellent  service.     Bearing  surfaces 
are  made  of  many  different  substances  depending  largely  upon 
the  class  of  service  for  which  the  bearing  is  intended.     The  fol- 
lowing is  a  list  of  some  of  the  materials  that  are  used  for  bearing 
surfaces:  babbitt  metal;  various  grades  of  bronzes;  cast  iron; 
mild,  case-hardened,  and  tempered  steel;  wood;  fiber  graphite. 

The  main  requirements  for  a  good  bearing  metal  are  the  follow- 
ing: (1)  It  should  possess  sufficient  strength  to  prevent  squeezing 
out  of  the  bearing  when  subjected  to  a  load.  (2)  It  should  not 
heat  rapidly  and  should  have  a  high  melting  point.  (3)  It 


ART.  355] 


BEARING  MATERIALS 


515 


should  be  able  to  resist  abrasion  but  should  not  score  the  journal. 
(4)  It  should  be  uniform  in  texture  and  possess  a  low  coefficient 
of  friction. 

(a)  Babbitt  metal. — Babbitt  metal  is  used  more  extensively 
than  any  other  bearing  metal.  One  reason  is  that  the  metal  is 
easily  melted  in  a  common  ladle  and  poured  into  the  bearing. 
Babbitt  bearings  require  an  outer  shell  to  which  the  metal  is 
anchored.  Generally  the  shell  is  made  of  cast  iron  although  steel 
casting  and  bronze  are  sometimes  used.  Shells  made  of  bronze 
have  the  advantage  that  in  case  the  babbitt  metal  melts  and 
runs  out  of  the  bearing,  the  journal  will  not  be  damaged  so  read- 
ily. The  babbitt  lining  is  made  about  %g  inch  thick  in  small 
bearings  and  from  %  to  %  mch  thick  in  large  bearings.  To  pre- 
vent rotation  of  the  babbitt  lining,  the  shell  must  be  provided 
with  some  form  of  anchor.  These  anchors  may  consist  of  dove- 


FIG.  289. 

tailed  grooves  as  shown  in  Fig.  289,  or  cored  or  drilled  holes  into 
which  the  babbitt  may  flow  when  the  bearing  liner  is  cast. 

Babbitt  metals  having  various  degrees  of  hardness  are  in  use. 
A  so-called  hard  babbitt  is  suitable  for  bearings  subjected  to 
heavy  pressure  or  severe  shock,  while  a  soft  babbitt  is  better 
adapted  to  a  light  load  and  high  speed.  Babbitt  metal  is  used 
for  the  main  bearings  of  engines  and  air  compressors,  on. steam 
turbines,  centrifugal  pumps  and  blowers,  motors  and  generators, 
in  wood-working  machinery,  in  bearings  for  line-  and  counter- 
shaft, and  in  many  machine  bearings  of  the  split  type.  For  the 
compositions  of  several  grades  of  babbitt  metals  see  Art.  51, 
Chapter  II. 

(b)  Bronzes. — Next  to  babbitt  metal,  bronze  is  considered  the 
most  important  bearing  material.  It  is  commonly  used  in  the 
form  of  a  one-piece  bushing  forced  under  pressure  into  the  shell 
or  framework  of  the  bearing.  Frequently  the  bushing  is  split 
into  halves  each  of  which  is  fastened  by  suitable  means  to  a  part 


516  PROVISIONS  FOR  LUBRICATION          [CHAP.  XIX 

of  the  bearing  shell.  The  thickness  of  these  bushings  varies 
with  the  diameter  and  the  length  of  the  journal.  There  are 
upon  the  market  a  large  number  of  different  kinds  of  bronzes, 
many  of  which  are  giving  excellent  service.  For  the  composition 
and  other  information  pertaining  to  several  grades  of  commercial 
bronzes  see  Art.  48,  Chapter  II. 

(c)  Cast  iron. — Cast-iron  bearings  running  with  steel  journals 
have  met  with  considerable  success  and  eminent  engineers  have 
advocated  their  use,  claiming  that  the  surface  will  in  a  short 
time  wear  to  a  glassy  finish  and  run  with  very  little  friction. 
However,  if  for  any  reason  lubrication  fails  and  heating  begins, 
the  result  is  liable  to  be  either  serious  injury  or  total  destruction 
to  both  bearing  and  journal.  Several  machine-tool  builders 
use  cast-iron  bearings  that  are  constantly  flooded  with  oil  and 
they  experience  no  bearing  troubles.  In  general  it  may  be  said 
that  cast-iron  bearings  will  prove  satisfactory  when  the  pressure 
and  speed  coming  upon  the  bearing  are  not  excessive  and  where 
sufficient  lubrication  is  insured. 

356.  Provisions  for  Lubrication. — The  object  of  any  system  of 
lubrication  is  to  form  and  maintain  a  uniform  film  of  oil  between 
the  journal  and  its  bearing.  By  the  term  system  of  lubrication 
is  meant  the  method  used  for  bringing  the  lubricant  to  the 
bearing  and  its  distribution  in  the  bearing.  To  distribute  the 
oil  and  assist  in  the  formation  of  a  uniform  oil  film,  the  bearing  is 
generally  provided  with  a  series  of  oil  grooves.  These  grooves 
should  start  at  the  point  of  supply  and  lead  diagonally  outward 
in  the  direction  of  rotation.  For  journals  rotating  in  either 
direction,  the  bearing  is  provided  with  a  symmetrical  arrange- 
ment of  grooves.  To  insure  the  formation  of  the  oil  film,  the 
edges  of  the  oil  grooves  must  be  bevelled  or  rounded  off.  The 
lubricant  is  delivered  to  the  bearing  or  to  the  journal  in  various 
ways  among  which  are  the  following:  (1)  Drop-feed  lubrication; 
(2)  wick  lubrication;  (3)  saturated-pad  lubrication ;  (4)  chain  or 
ring  lubrication;  (5)  flooded  lubrication;  (6)  forced  lubrication; 
(7)  grease  lubrication. 

(a)  Drop-feed  lubrication. — The  most  common  method  of  oiling 
a  bearing  is  by  means  of  the  drop-feed  method.  In  its  simplest 
form  it  consists  of  an  open  hole  in  the  bearing  through  which  oil 
is  introduced.  In  many  cases  the  hole  is  tapped  to  receive  a 
closed  oil  cup,  thus  preventing  dirt  and  grit  from  entering  the 
bearing. 


ART.  356] 


PROVISIONS  FOR  LUBRICATION 


517 


(b)  Wick  lubrication. — In  bearings  used  on  line-  and  counter- 
shafts and  occasionally  in  machinery,  the  oil  is  transferred  by 
capillary  action  from  a  small  reservoir  in  the  cap  to  the  bearing 
surfaces  by  means  of  a  wick  as  shown  in  Fig.  290.  This  method 
of  lubrication  is  satisfactory  when  the  bearing  pressures  and  the 
speed  are  not  excessive. 


FIG.  290. 

(c)  Saturated-pad  lubrication. — An  effective  way  of  lubricating 
line-  and  countershaft  bearings  is  by  means  of  wooden  blocks  con- 
taining a  series  of  saw-cuts  through  which  the  oil  rises.  The 
blocks,  generally  two  in  number,  are  located  in  the  lower  half  of 
the  bearing  and  are  held  in  contact  with  the  shaft  by  means  of 
springs.  The  lower  ends  of  these  blocks  project  into  the  oil 


FIG.  291. 

reservoir,  thus  permitting  the  lubricant  to  rise  from  the  reservoir 
to  the  shaft  by  means  of  capillary  action. 

(d)  Ring  or  chain  lubrication. — Ring  or  chain  lubrication  is 
considered  one  of  the  best  methods  of  supplying  a  bearing  with 
oil.  It  is  used  on  bearings  for  all  classes  of  machinery.  An 
application  of  a  ring  oiler  to  a  line-  and  countershaft  bearing  is 
shown  in  Fig.  291,  and  in  Fig.  296,  297,  300,  310  and  311  are 
shown  various  designs  such  as  are  used  on  machine  tools,  cen- 


518  PROVISIONS  FOR  LUBRICATION          [CHAP.  XIX 

trifugal  pumps,  etc.  The  quantity  of  oil  delivered  to  the  bearing 
by  a  ring  depends  upon  the  size  and  speed  of  the  ring  and  upon 
the  viscosity  of  the  oil.  The  diameter  of  the  oil  ring  should  be 
made  approximately  double  the  diameter  of  the  shaft,  and  the 
ring  may  be  made  solid  or  split.  The  former  construction  is 
used  for  small  bearings  and  the  latter  for  larger  bearings.  An 
inspection  of  a  considerable  number  of  ring-oiling  bearings  used 
on  line-shafts,  electrical  machinery,  and  centrifugal  pumps  seems 
to  indicate  that  an  oil  ring  -cannot  be  expected  to  supply  proper 
lubrication  over  a  length  of  bearing  exceeding  approximately 
4  inches  on  each  side  of  the  ring.  In  electrical  machinery  the 
rings  are  usually  made  of  brass  or  bronze  in  order  to  avoid  mag- 
netic difficulties.  In  general  the  rings  should  be  perfectly  round, 
they  should  have  no  sharp  corners,  and  they  should  be  well 
balanced. 

On  the  main  bearings  of  high-speed  engines  a  form  of  bearing 
similar  to  the  ring-oiling  type  is  occasionally  used,  but  in  place  of 
the  ring  a  sash  chain  is  used. 

(e)  Flooded  lubrication. — In  flooded  lubrication  the  oil  is  sup- 
plied to  the  bearing  by  means  of  a  pump  or  from  an  overhead 
reservoir,  but  at  practically  no  pressure,  This  system  has  been 
used  to  some  extent  on  machine  tools. 

(/)  Forced  lubrication. — In  forced  lubrication  the  oil  is  supplied 
to  the  bearing  at  a  considerable  pressure  by  means  of  a  pump. 
Generally  the  oil  pressure  varies  from  15  to  25  pounds  per  square 
inch;  however,  the  pressure  may  run  up  to  600  pounds  per  square 
inch  as  in  the  case  of  the  step  bearing  used  on  Curtis  vertical 
steam  turbines. 

(g)  Grease  lubrication. — Grease  lubrication  is  well  adapted  for 
use  on  bearings  subjected  to  heavy  pressures  and  in  which  the 
speeds  are  relatively  low.  Grease  is  introduced  into  the  bearing 
by  any  one  of  the  various  forms  of  grease  cups  obtainable  on  the 
market. 

Very  few  of  the  systems  of  lubrication  discussed  above  produce 
a  perfect  oil  film.  According  to  Axel  K.  Pederson,  analytical 
expert  of  the  General  Electric  Co.,  the  various  systems  given 
above  may  be  arranged  into  the  following  three  classes: 

1.  Those  systems  which  produce  an  imperfect  oil  film;  for  ex- 
ample, drop-feed,  wick,  and  grease  lubrication. 

2.  Those  systems  which  produce  a  semi-perfect  oil  film,  for  ex- 
ample, saturated-pad  and  ring  or  chain  lubrication. 


ART.  357] 


ADJUSTMENTS  FOR  WEAR 


519 


3.  Those  systems  which  produce  a  perfect  oil  film;  for  example, 
flooded  and  forced  lubrication. 

357.  Adjustments  for  Wear. — (a)  Split  bearing. — In  the  major- 
ity of  bearings  some  means  of  taking  up  wear  must  be  provided. 
The  adjustment  for  wear  may  be  made  in  various  ways,  but  the 


FIG.  292. 


most  common  method  is  by  the  use  of  a  split  bearing  the  parts 
of  which  are  bolted  together.  The  wear  is  taken  up  by  simply 
removing  some  of  the  metal  or  paper  shims  and  tightening  the 
bolts  in  the  bearing  cap.  In  split  bearings  the  line  of  division 


FIG.  293. 

should  be  made  with  an  offset  as  shown  in  Fig.  292  and  293  for 
two  reasons:  (1)  When  made  with  an  offset,  the  cap  will  pre- 
vent the  bearing  under  pressure  from  springing  together  at  the 
sides  and  gripping  the  shaft.  (2)  The  offset  will,  to  a  certain 
extent,  prevent  the  escape  of  the  lubricant. 


520 


ADJUSTMENTS  FOR  WEAR 


[CHAP.  XIX 


(b)  Four-part  bearing. — The  main  bearings  of  steam  and  gas 
engines  are  generally  of  the  four-part  type  similar  to  the  design 
shown  in  Fig.  294.  The  babbitt-lined  side  shells  are  provided 
with  adjusting  wedges  which  extend  the  full  length  of  the  bearing. 
The  bottom  shell  is  also  lined  with  babbitt  metal  and  rests  in  a 
spherical  seat  in  the  engine  frame,  thus  keeping  the  shaft  in  good 
alignment  at  all  times.  By  raising  the  shaft  sufficiently  to  relieve 
the  bearing  of  its  load,  the  bottom  shell  may  be  rolled  out  and 
inspected. 

As  shown  in  Fig.  294,  the  bearing  cap  is  of  heavy  construction 
and  is  not  babbitted  the  entire  length  of  the  bearing,  but  merely 
for  a  short  distance  at  each  end.  The  cap  is  placed  over  the 


FIG.  294. 

jaws  of  the  engine  frame  with  a  driving  fit.  It  is  evident  that 
a  four-part  bearing  permits  making  adjustments  for  wear  in  a 
more  nearly  correct  manner  than  is  possible  with  a  common  split 
bearing;  hence  it  is  well  adapted  for  installation  where  the  line 
of  action  of  the  resultant  bearing  pressure  changes  with  the  rota- 
tion of  the  shaft. 

(c)  Solid  bearing. — No  doubt  the  simplest  form  of  bearing  is 
that  known  as  the  solid  type,  designs  of  which  are  shown  in  Figs. 
295  to  298  inclusive.  The  solid  bearing  has  no  provision  for 
taking  up  wear  except  by  removing  the  worn-out  bushing  or 
liner  and  replacing  it  with  a  new  one.  The  bronze  bushed  bear- 
ings shown  in  Figs.  295  and  296  have  been  used  successfully  on 
heavy  machine  tools.  They  have  ample  provisions  for  lubrica- 
tion, but  none  for  wear  except  by  renewal  of  the  bushing.  Such 
bushings  are  replaced  very  readily  at  a  small  cost.  In  Fig.  297 
is  shown  another  design  of  ring  oiling  solid  bearing  consisting  of  a 


ART.  357] 


ADJUSTMENTS  FOR  WEAR 


521 


cast-iron  shell  lined  with  babbitt  metal.  This  type  of  bearing 
has  been  found  to  give  good  service  on  the  small  and  medium 
sizes  of  centrifugal  pumps.  The  shell  and  brass  oil  ring  are  fitted 
into  a  suitable  housing;  the  shell  is  held  in  place  by  a  special 
headless  screw  projecting  into  the  hole  shown  in  the  figure. 


FIG.  295. 

The  design  of  a  solid  bearing  shown  in  Fig.  298  is  used  in  places 
where  the  pressure  upon  the  bearing  is  always  in  the  same  direc- 
tion, as  for  example  on  the  shaft  used  for  supporting  the  overhead 
sheaves  of  an  elevator.  As  shown,  merely  the  lower  half  of  the 


FIG.  296. 

bearing,  which  in  this  case  takes  the  entire  pressure,  is  lined  with 
babbitt  metal.  The  central  part  of  the  bearing  shell  is  made 
spherical  so  that  it  will  fit  into  the  spherical  seat  in  the  pedestal, 
thus  keeping  the  shaft  in  proper  alignment. 

A  design  of  a  solid  bearing  used  on  the  spindles  of  heavy  milling 
machines  made  by  the  Ingersoll  Milling  Machine  Co.  is  shown  in 


522 


ADJUSTMENTS  FOR  WEAR 


[CHAP.  XIX 


Fig.  299.  The  conical  journal  of  the  spindle  a  is  fitted  with  a 
bronze  bushing  b,  the  latter  being  forced  in  the  sleeve  g.  Some- 
where near  the  middle  of  its  length,  the  spindle  has  keyed  to  it  a 
removal  conical  journal  c.  The  latter  fits  into  the  conical  bronze 
bushing  /,  which  is  forced  into  the  sliding  sleeve  g.  Practically 
all  of  the  wear  comes  upon  the  bearing  b  and  may  be  taken  up  by 


FIG.  297. 

means  of  the  adjusting  nut  e  and  the  special  washer  provided 
between  the  end  of  6  and  the  enlarged  head  of  the  spindle.  Due 
to  the  use  of  the  conical  bearing,  the  alignment  of  the  spindle  is 
not  disturbed  by  an  appreciable  amount  when  an  adjustment  for 
wear  is  made. 

On  the  spindles  of  certain  machine  tools  the  bearings  are  made 
with  a  bronze  bushing  having  a  straight  bore  and  is  turned  conical 


FIG.  298. 

on  the  outside  as  shown  in  Fig.  300.  The  bushing  is  threaded  at 
each  end  and  is  provided  with  a  slit  extending  through  the  entire 
length.  It  is  evident  that  this  bearing  may  readily  be  adjusted 
for  wear  by  means  of  the  adjusting  nuts  at  the  ends  of  the  bearing. 
The  oil  ring  and  oil  reservoir  provided  in  the  framework  of  the 
bearing  insure  proper  lubrication  of  the  bearing  at  all  times. 


ART.  357] 


ADJUSTMENTS  FOR  WEAR 


523 


(d)  Connecting-rod  bearings. — The  bearings  used  on  connecting 
rods  differ  somewhat  from  those  discussed  in  preceding  para- 
graphs. In  Figs.  301  to  303  inclusive  are  shown  three  designs  that 
have  proven  satisfactory.  The  first  two  are  used  on  the  crankpin 
end  of  the  rod  while  the  third  is  intended  for  the  crosshead  end, 


FIG.  299. 


although  a  similar  design  is  frequently  used  for  the  crankpin 
end.  In  all  three  cases  the  adjustments  for  wear  are  made  by 
means  of  a  wedge  and  suitable  cap  screws. 

The  design  shown  in  Fig.  301  consists  of  two  half  bearings 


FIG.  300. 


around  which  a  steel  stirrup  or  strap  is  placed,  the  latter  being 
fastened  rigidly  to  the  rod  end  by  two  through  bolts.  The  ad- 
justing wedge  with  its  screws  is  located  between  the  strap  and 
the  front  half  of  the  bearing.  Taking  up  wear  by  means  of  this 


524 


ADJUSTMENTS  FOR  WEAR 


[CHAP.  XIX 


d 


FIG.  301. 


FIG.  302. 


ART.  357] 


ADJUSTMENTS  FOR  WEAR 


525 


wedge  tends  to  shorten  the  rod,  hence  the  bearing  at  the  other  end 
of  the  rod  should  be  equipped, with  an  adjustment  which  tends  to 
counteract  the  former,  thus  maintaining  a  constant  distance 
between  the  two  bearings.  For  economy  of  material  the  two 
halves  of  the  bearing  are  made  of  steel  casting  lined  with  babbitt 
metal.  Sometimes  brass  is  used  in  place  of  the  steel  casting. 

In  Fig.  302  is  shown  an  open  rod  end  into  which  are  fitted  the 
two  halves  of  the  bearing;  one  of  these  halves  is  movable  and 
the  other  is  fastened  rigidly  to  the  rod  by  a  through  bolt.  The 
adjusting  wedge  and  screws  are  located  between  the  back  bearing 


f—  C7~ 

^ZLj 

r 

f 

/T\ 

\\j 

\ 

-^          -K 

1 

1 

J 

FIG.  303. 

and  the  rod,  thus  the  tendency  is  to  lengthen  the  rod  when  the 
wear  is  taken  up.  As  in  the  design  shown  in  Fig.  301,  the  two 
halves  of  the  bearing  are  made  of  steel  casting  lined  with  babbitt 
metal. 

The  design  shown  in  Fig.  303  is  called  a  closed-rod  end.  The 
adjustments  for  wear  are  made  in  the  same  manner  as  in  the 
preceding  designs.  It  is  evident  from  the  figure  that  taking 
up  wear  tends  to  lengthen  the  connecting  rod,  hence  this  design 
would  be  a  proper  one  to  use  in  connection  with  that  shown  in 
Fig.  301  since  the  length  of  the  rod  would  remain  practically  a 
constant.  The  two  parts  of  the  bearing  used  with  the  closed- 


526 


ADJUSTMENTS  FOR  ALIGNMENT          [CHAP.  XIX 


rod  end  of  Fig.  303  are  generally  made  of  bronze  though  occasion- 
ally babbitt-lined  bearings  are  used. 

358.  Adjustments  for  Alignment. — In  addition  to  provisions 
for  taking  up  wear,  many  bearings  are  provided  with  means  for 
aligning  the  shaft.  Bearings  that  are  out  of  line  tend  to  heat 
and  produce  wear.  Some  of  the  bearings  discussed  in  Art. 
357  meet  the  provisions  for  alignment  by  having  the  bearing 
divided  into  parts  that  can  be  adjusted  vertically  or  horizontally, 
while  others  are  provided  with  spherical  seats  thus  making  them 
self-aligning.  In  many  cases  the  bearing  and  its  housing  are 


FIG.  304. 

mounted  on  supports  which  permit  the  adjustments  necessary 
to  line  up  the  shaft.  The  horizontal  adjustment  in  such  cases 
is  generally  provided  for  by  elongating  the  holes  through  which 
the  housing  is  bolted  to  the  support. 

Hangers. — For  lining  up  the  bearings  of  line-  and  countershafts 
various  forms  of  hangers  are  used.  As  shown  in  Figs.  289  to  291 
inclusive,  line-shaft  bearings  are  made  in  two  parts  each  of  which 
is  provided  with  a  spherical  seat  which  fits  into  a  corresponding 
seat  on  the  sockets  of  the  hangers.  A  design  of  a  cast-iron  single- 
brace  ball  and  socket  drop  hanger  is  shown  in  Fig.  304.  From 
this  figure  it  is  evident  that  the  two-ball  seated  sockets  provide 
the  vertical  adjustment  while  the  slotted  holes  in  the  supporting 


ART.  359] 


BEARING  PRESSURES 


527 


flanges  of  the  hanger  take  care  of  the  horizontal  adjustment. 
When  greater  rigidity  is  required  than  is  furnished  by  the  hanger 
shown  in  Fig.  304,  a  double-brace  design  similar  to  that  repre- 
sented in  Fig.  305  is  used.  However,  the  hanger  shown  in  the 
latter  figure  is  made  entirely  of  pressed  steel,  the  parts  being 
riveted  or  bolted  together  as  shown. 
Set  screws  are  used  for  giving  the 
desired  adjustments. 


DESIGN  OF  BEARINGS  AND 
JOURNALS 

359.  Bearing  Pressures.  —  In  order 
to  maintain  an  oil  film  between  the 
journal  and  its  bearing,  the  pressure 
must  not  exceed  the  so-called  critical 
pressure,  by  which  is  meant  the 
limiting  pressure  at  which  a  perfect 
film  between  the  journal  and  the 
bearing  is  maintained.  This  pres- 
sure depends  upon  the  speed  of  the 
journal,  the  viscosity  of  the  oil,  the 
temperature  of  the  bearing,  the  close- 
ness of  the  fit  between  the  journal 
and  its  bearing,  and  the  degree  of 
finish  given  to  the  surfaces  in  con- 
tact. As  yet  no  test  results  are. 
available  to  show  the  relation  exist- 

ing between  the  pressure,  viscosity,  and  temperature.  According 
to  H.  F.  Moore,  the  relation  existing  between  the  critical  pres- 
sure pc  and  the  speed  of  the  journal  is  given  by  the  following 
formula  : 

pc  =  7.47  VV,  (561) 

in  which.  V  denotes  the  peripheral  speed  of  the  journal  in  feet 
per  minute.  The  Moore  formula  is  based  upon  the  results 
obtained  from  a  series  of  experiments  on  a  steel  journal  running 
on  a  white  metal  bearing.  The  pressure  carried  on  the  bearing 
varied  from  10  to  80  pounds  per  square  inch  of  projected  area, 
and  the  speed  did  not  exceed  140  feet  per  minute. 

The  following  formula  for  the  permissible  bearing  pressure 
based  on  Stribeck's  results  is  taken  from  Smith  and  Marx's 


FIG.  305. 


528 


BEARING  PRESSURES 


[CHAP.  XIX 


"  Machine   Design,"    and   is   recommended   for   use   when   the 
speed  does  not  exceed  500  feet  per  minute: 


P 


TABLE  96.  —  ALLOWABLE  BEARING  PRESSURES 


(562) 


Type  of 

bearing 

Pressure  p 

Punching  and  shearing 
machinery 

Main  journals 
Crankpins 

2,000-3,000 
5,000-8,000 

Engine  crankpins 

High  speed 
Low  speed 
Locomotives 
Marine 
Air  compressors,  center  crank 
Auto  gas  engines 

250-600 
850-1,500 
1,500-1,700 
400-600 
250-500 
350-450 

Engine  crosshead  pins 

High  speed 
Low  speed 
Locomotives 
Marine 
Air  compressors,  center  crank 
Auto  gas  engine 

900-1,700 
1,000-1,800 
3,000-4,000 
1,000-1,500 
400-800 
800-1,000 

Engine  main  bearings 

High  speed 
Low  speed 
Marine 
Air  compressors,  center  crank 
Auto  gas  engine 

180-240 
160-220 
200-400 
150-250 
350-400 

Locomotive  driving  journals 

Passenger 
Freight 
Switching 

190 
200 
220 

Engine  crossheads 

Stationary 
Marine 

25-40 
50-100 

Car  journals 

300-500 

Motors  and  generators  

40-80 

Horizontal  steam  turbines 

40-60 

Eccentric  sheaves 

80-100 

Hoisting  machinery  shafting. 

70-90 

Propeller  shaft  thrust  bear- 
ings. 

Freight  steamer 
Passenger  steamer 
Large  naval  vessels 
Light  naval  vessels 

40-55 
60-80 
70-90 
110-130 

ART.  360]   RELATION  BETWEEN  LENGTH  -AND  DIAMETER   529 


For  speeds  exceeding  500  feet  per  minute,  the  same  authorities 
suggest  the  formula 

pc  =  30^/7  (563) 

In  L.  P.  Alford's  book  on  " Bearings"  is  given  a  chart  showing 
the  relation  between  the  maximum  safe  bearing  pressure  and  the 
rubbing  speed  for  perfect  film  lubrication.  This  chart  represents 
the  practice  of  the  General  Electric  Co.  in  designing  the  bearings 
used  on  motors  and  generators.  The  following  expression  gives 
values  of  the  maximum  safe  bearing  pressure  which  agree  very 
closely  with  those  obtained  from  the  chart. 

pm  =  15.5V/F  (564) 

In  addition  to  the  formulas  given  in  this  article,  the  allowable 
bearing  pressures,  in  pounds  per  square  inch  of  projected  area, 
contained  in  Table  96  will  serve  as  a  guide  in  designing  bearings 
and  journals.  These  values  are  based  upon  current  practice 
and  were  collected  from  various  sources. 

TABLE  97. — RELATION  BETWEEN  LENGTH  AND  DIAMETER  OF  BEARINGS 


Ratio  l/d 

Ratio  l/d 

Min. 

Max. 

Min. 

Max. 

Marine 

Main  bearing 

1.00 

1.50 

Steam  turbines 

2.0 

3.0 

engine 

Crankpin  bearing 

1.00 

1.50 

Generators  and  motors 

2.0 

3.0 

High-speed 

Main  bearing 
Crankpin  bearing 

2.00 
1.00 

3.00 
1.00 

Centrifugal  pumps 
Centrifugal  fans 

2.0 
2.0 

2.5 
3.0 

Crosshead  pin  bearing 

1.40 

1.60 

Machine  tools 

2.0 

4.0 

Slow-speed 

Main  bearing 
Crankpin  bearing 

1.75 
1.00 

2.25 
1.25 

Hoisting  drums 
Hoisting  sheaves  for  cranes 

1.5 
1.0 

2.0 
2.0 

Crosshead  pin  bearing 

1.20 

1.50 

Wood  working  machinery 

2.5 

4.0 

Stationary 

Main  bearing 
Crankpin  bearing 

2.00 
1.00 

2.50 
1.50 

Shaft 
hangers 

Rigid 

Self-adjusting 

2.5 
3.0 

3.0 
4.0 

f^           \\       A      '      K         " 

1    ^O 

i    7* 

ross   ea    pin    earing 

Pillow 

Plain 

2   5 

3  5 

Auto  gas 

Main  bearing 

1.00 

1.75 

blocks 

Ring-oiling 

4.0 

5.0 

engine 

Crankpin  bearing 

1.20 

1.40 

Crosshead  pin  bearing 

1.70 

2.25 

360..  Relation  between  Length  and  Diameter. — The  ratio  of 
the  length  of  a  bearing  to  its  diameter  is  fairly  well  established 
for  the  different  classes  of  machinery.  In  Table  97  are  given  the 
values  of  this  ratio  for  a  considerable  number  of  different  types 


530  RADIATING  CAPACITY  [CHAP.  XIX 

of  bearings,  the  majority  of  which  were  obtained  from  a  study 
of  actual  installations. 

361.  Radiating  Capacity  of  Bearings. — The  capacity  that  a 
bearing  has  for  radiating  the  heat  generated  by  the  friction  be- 
tween the  journal  and  its  bearing  depends  upon  the  mass  of 
metal  used   in  the  construction  of   the  bearing  and  upon  the 
condition  of  the  surrounding  air.     The  following  formula  due 
to  Axel  K.  Pedersen  may  be  used  for  determining  the  amount 
of  heat  carried  away: 

Q  -  ^+  ^  (565) 

in  which  Q  denotes  the  heat  radiating  capacity  of  a  bearing  ex- 
pressed in  foot-pounds  per  second  per  square  inch  of  projected 
area;  TQ  denotes  the  difference  between  the  temperature  of  the 
bearing  and  that  of  the  cooling  medium;  K  denotes  an  experi- 
mental constant  the  magnitude  of  which  depends  upon  the 
method  used  for  cooling  the  bearing.  The  following  values  of 
K  derived  by  Pedersen  from  Lasche's  and  the  General  Electric 
Co.'s  experiments  may  be  safely  used  in  designing  bearings: 

1.  For    bearings  of  light  construction  located  in  still  air — 
K  =  3,300. 

2.  For  bearings  of  heavy  construction  and  well  ventilated — 
K  =  1,860. 

3.  For    General    Electric    Co.'s    well-ventilated    bearings — 
K  =  1,150. 

362.  Coefficient  of  Friction. — The  coefficient  of  friction  be- 
tween the  bearing  and  its  journal  depends  upon  the  bearing 
pressure,  the  speed  of  the  journal,  the  temperature  of  the  bear- 
ing, the  specific  gravity  of  the  lubricant,  and  the  method  used 
for  lubricating  the  bearing.     The  laws  governing  the  coefficient 
of  friction  in  a  bearing  provided  with  a  limited  supply  of  lubri- 
cant are  generally  assumed  the  same  as  those  governing  ordinary 
sliding  friction.     However,  when  the  bearing  is  provided  with  a 
copious  supply  of  lubricant,  the  coefficient  of  friction  depends 
upon  the  laws  of  friction  in  a  fluid,  that  is,  the  resistance  the 
lubricant  offers  against  shearing. 

In  an  article  entitled  " Bearing  Design  Constants"  which 
appeared  in  Power,  Feb.  22,  1916,  Mr.  Louis  Illmer  gives  several 
formulas  for  the  coefficient  of  friction  which  are  based  upon  the 


ART.  362]  COEFFICIENT  OF  FRICTION  531 

experimental  researches  of  Tower,  Lasche,  Thomas,  Maurer 
and  Kelso.  The  coefficient  of  friction  according  to  the  Tower 
tests  is  given  by  the  expression 


2     tV 

=  v\r 


(566) 


in  which  p  denotes  the  bearing  pressure  in  pounds  per  square 
inch  of  projected  area;  V  the  speed  of  the  journal  in  feet  per 
minute;  T  the  virtual  temperature  head  of  the  oil,  which  may 
be  assumed  as  the  temperature  of  the  bearing  less  60°.  Ac- 
cording to  Illmer  this  formula  is  applicable  to  bearings  having  a 
pressure  range  of  100  to  500  pounds  per  square  inch  of  projected 
area,  and  in  which  the  speed  does  not  exceed  500  feet  per 
minute. 

The  Lasche  experiments  were  made  on  a  steel  journal  running 
in  a  ring  oiling  babbitt  lined  bearing,  and  the  results  obtained 
lead  to  the  following  expression  for  the  coefficient  of  friction: 

(567) 


This  formula  is  applicable  to  bearings  subjected  to  pressures 
of  15  to  225  pounds  per  square  inch  of  projected  area,  and  in 
which  the  speed  may  range  from  500  to  3,500  feet  per  minute. 
The  temperature  of  the  bearing  may  vary  from  85°  to  210°F. 

From  the  results  of  experiments  made  by  Thomas,  Maurer  and 
Kelso  on  babbitt-lined  hanger  bearings,  Illmer  derived  the  fol- 
lowing formula  for  the  coefficient  of  friction: 

Vv 

M  =  --  7=  (568) 


The  use  of  this  formula  is  limited  to  bearings  in  which  the  pres- 
sures vary  from  33  to  100  pounds  per  square  inch  of  projected 
area,  and  in  which  the  speed  of  the  journal  ranges  from  100  to 
300  feet  per  minute. 

The  experiments  of  Lasche,'  as  well  as  some  made  at  Cornell 
LTniversity,  seem  to  indicate  that  the  coefficient  of  friction  is 
practically  independent  of  the  speed  when  the  latter  exceeds 
500  feet  per  minute.  Upon  this  assumption,  (568)  may  be  sim- 
plified by  substituting  for  V  the  critical  value  500,  whence 


532  DESIGN  FORMULAS  [CHAP.  XIX 

Equation  (569)  proposed  by  Illmer,  gives  values  of  the  coefficient 
of  friction  that  may  reasonably  be  expected  in  the  operation  of 
well-designed  bearings  lined  with  babbitt  metal  and  lubricated 
with  a  generous  supply  of  mineral  engine  oil. 

Mr.  William  Knight,  in  the  American  Machinist  of  Nov.  16, 
1916,  suggests  that  (569)  be  modified  by  introducing  in  the  nu- 
merator a  factor  s  denoting  the  specific  gravity  of  the  oil  when 
compared  to  water.  Thus  the  revised  form  of  (569)  becomes 


Knight  bases  his  suggestion  upon  an  investigation  of  the  results 
obtained  by  A.  L.  Westcott  from  a  series  of  tests  made  at  the 
University  of  Missouri  on  greases  and  oils.  Furthermore  (570) 
gives  values  of  n  that  agree  fairly  well  with  the  results  obtained 
by  Lasche  for  pressures  between  120  and  240  pounds  per  square 
inch. 

363.  Design  Formulas.  —  Having  given  the  ratio  between  the 
length  of  the  bearing  and  its  diameter,  we  may  readily  develop 
working  formulas  for  the  diameter  of  the  bearing  in  terms  of  the 
load  P,  the  revolutions  per  minute,.  and  certain  constants.  The 
resultant  formulas  will  be  based  upon  equations  (562)  and  (563)  ; 
hence  they  will  only  apply  to  bearings  receiving  a  copious  supply 
of  lubricant  and  to  those  in  which  the  speed  remains  within  the 
range  given  in  Art.  359.  For  a  bearing  having  a  diameter  d  and 
length  I  and  subjected  to  a  total  pressure  P,  the  pressure  per 
square  inch  of  projected  area  is 

P  =  £,  (571) 

in  which  c  denotes  the  ratio  of  I  to  d.  Equating  the  value  of  p 
to  the  limiting  pressure  given  by  (562),  we  obtain 

P  =  10  cd*Vv  (572) 

Introducing  the  value  of  V  in  terms  of  d  and  N,  the  number  of 
revolutions  per  minute  of  the  journal,  we  obtain  the  following  ex- 
pression for  d  for  speeds  below  500  feet  per  minute: 

d  =  0.52^1  (573) 

By  a  similar  procedure,  using  (563)  in  place  of  (562),  we  obtain 


ART.  364]  TEMPERATURE  OF  BEARINGS  533 

the  following  formula  for  d  for  speeds  exceeding  500  feet  per 
minute: 

P3 


=  0.282  !/ 


(574) 

By  means  of  (573)  or  (574),  whichever  applies  to  the  problem 
under  discussion,  the  diameter  of  the  bearing  may  be  calculated. 
Knowing  d,  the  length  of  the  bearing  may  be  determined  since 
I  =  cd.  Furthermore,  the  magnitude  of  p  may  be  determined  by 
means  of  (571). 

364.  Temperature  of  Bearings. — Frequently  it  is  desirable  to 
determine  the  probable  temperature  of  the  bearing  due  to  the 
heat  generated.  If  the  temperature  becomes  too  high  the  oil  is 
liable  to  lose  its  lubricating  qualities,  hence  it  may  be  necessary 
to  redesign  the  bearing  or  resort  to  artificial  cooling.  The  work 
of  friction  expressed  in  foot-pounds  per  second  per  square  inch 
of  projected  area  is 

Wf  =  ^  (575) 

and  this  must  equal  the  quantity  of  heat  radiated  or  carried  away 
as  expressed  by  (565) ;  hence 

WV  _  (T0  +  33V 
60   "  K 

from  which  the  limiting  speed  of  the  bearing  for  a  given  final 

temperature  is 

fin 
V  =  ~  (To  +  33)°  (577) 

Equation  (577)  may  also  be  used  to  calculate  the  probable 
temperature  of  a  well-lubricated  bearing  running  under  given 
conditions  of  load  and  speed.  To  determine  this  temperature 
the  following  method  of  procedure  is  suggested:  From  (568)  or 
(570),  depending  upon  the  speed,  determine  the  value  of  JJL  in 
terms  of  T.  Substitute  this  value  of  /*  as  well  as  the  magnitudes 
of  p,  V,  and  K  in  (577)  and  determine  the  probable  temperature 
of  the  bearing.  The  maximum  temperature  of  a  bearing  depends 
upon  the  lubricant  used,  and  since  bearing  oils  begin  to  show  signs 
of  losing  their  lubricating  qualities  at  a  temperature  of  approxi- 
mately 250°F.  it  is  considered  good  practice  to  limit  the  maximum 
temperature,  as  determined  by  (577),  to  180°F. 


534  DESIGN  OF  JOURNALS  [CHAP.  XIX 

365.  Strength  and  Stiffness  of  Journals.  —  In  the  majority  of 
cases  the  journal  is  integral  with  and  forms  a  part  of  the  shaft, 
the  diameter  of  which  has  been  calculated  according  to  the 
methods  given  in  Chapter  XVIII.  The  dimensions  of  the 
journals  of  important  shafts  are  not  generally  based  on  calcula- 
tions for  strength  and  stiffness  but  on  the  liability  of  heating, 
that  is,  the  conditions  which  govern  the  oil  supply.  However,  the 
stresses  in  a  journal  should  always  be  investigated  in  order  to 
make  sure  that  the  dimensions  are  ample  so  far  as  strength  and 
rigidity  are  concerned. 

(a)  Strength  of  end  journals.  —  End  journals  are  generally  con- 
sidered cantilever  beams  loaded  uniformly.  Equating  the  bend- 
ing moment  to  the  moment  of  resistance  and  solving  for  the  diam- 
eter d,  we  obtain 


d  =  1.72  (578) 

Having  given  the  dimensions  of  the  end  journal,  and  the  load 
coming  upon  it,  the  magnitude  of  the  stress  may  be  determined 
by  (578),  or  by  means  of  the  formula 

S  =  5.1  pc2,  (579) 

in  which  p  and  c  have  the  same  meaning  as  assigned  to  them  in 
Art.  363.  The  working  stress  £,  due  to  the  fatigue  of  the  mate- 
rial, should  not  exceed  4,000  to  5,000  pounds  per  square  inch. 

(6)  Stiffness  of  end  journals.  —  In  designing  journals  the  ques- 
tion of  stiffness  is  an  important  one,  and  should  be  given  the 
proper  consideration.  For  an  end  journal  loaded  uniformly,  the 
deflection  A  is  calculated  by  the  formulas 

2.55  PI* 

(580) 


whence 

(581) 


For  common  end  journals  good  engineering  practice  dictates  that 
the  value  of  A  should  not  exceed  0.01  of  an  inch. 

366.  Design  of  Bearing  Caps  and  Bolts. — The  cap  of  a  bearing 
should  never  be  subjected  to  a  heavy  load;  however,  there  are 
cases  in  which  the  circumstances  are  such  that  a  considerable 
pressure  comes  upon  the  cap.  In  such  cases  the  cap  is  generally 


ART.  366] 


DESIGNS  OF  CAPS  AND  BOLTS 


535 


regarded  as  a  beam  supported  by  the  holding  down  bolts  or 
screws  and  loaded  at  the  center,  as  shown  in  Fig.  306.  As  in 
the  journal,  the  cap  should  be  investigated  for  both  strength  and 
stiffness. 

(a)  Strength  of  cap. — Assuming  the  dimensions  of  the  cap  as 
represented  in  Fig.  306,  we  obtain  the  following  expression  for 
the  thickness  6,  by  equating  the  bending  moment  to  the  moment 
of  resistance: 

/O     7-»_ 

(582) 


(b)  Stiffness  of  cap. — In  order  that  the  cap  will  have  ample 
rigidity,  the  thickness  b  should  be  calculated  by  the  following 


FIG.  306. 

expression  based  upon  the  formula  for  the  deflection  A  of  a  simple 
beam  loaded  as  shown  in  Fig.  306: 

P  (583) 


6-  =  0.63  a 

For  the  cap  of  a  common  end  journal  or  a  marine  end  connecting 
rod,  good  engineering  practice  limits  the  deflection  A  to  0.01  of 
an  inch. 

(c)  Holding-down  bolts. — The  bolts,  screws,  or  studs  that  are 
used  for  holding  down  the  cap  are  generally  assumed  to  be  sub- 
jected to  a  simple  tension,  and  as  a  rule  each  bolt  is  designed  for 

4P 
a  load  equivalent  to  -~— ,  in  which  n  denotes  the  number  of  bolts 

used  for  holding  down  the  cap. 

367.  Work  Lost  Due  to  the  Friction  on  a  Cylindrical  Journal. 

— With  our  present  state  of  knowledge  of  the  subject  of  friction, 
we  are  unable  to  determine  a  correct  expression  for  the  work 
lost  due  to  journal  friction.  In  deriving  an  expression  for  the 
moment  of  journal  friction,  it  is  generally  assumed  that  the  coef- 
ficient of  friction  is  constant  for  a  given  speed  and  further  that 
the  pressure  between  the  surfaces  in  contact  is  uniformly  dis- 


536  FRICTION  LOSSES  IN  BEARINGS          [CHAP.  XIX 

tributed,  or  that  the  wear  of  the  journal  and  its  bearing  is  uni- 
form and  proportional  to  the  work  of  friction.  The  assumption 
of  uniform  distribution  of  pressure  is  hardly  warranted  in  the 
case  of  a  "worn-in"  journal  and  bearing;  but  for  a  new  journal 
and  bearing  having  perfect  contact  over  the  entire  bearing  sur- 
face, it  is  probable  that  the  pressure  is  uniformly  distributed. 
In  the  following  analysis,  formulas  based  on  both  assumptions 
will  be  derived. 

(a)  Pressure  uniformly  distributed. — Assuming  that  the  pres- 
sure between  the  journal  and  its  bearing  is  uniformly  distributed 
over  the  contact  surface,  the  intensity  of  pressure  p  is  equal  to 
the  load  P  on  the  journal  divided  by  the  projected  area  of  the 
journal.  This  may  be  shown  as  follows: 

The  pressure  on  a  longitudinal  strip  of  width  ds  and  length  7 
is  plds.  Let  the  direction  of  the  pressure  p  make  an  angle  6 
with  the  vertical  center  line  of  the  journal,  and  assume  that  the 
load  P  acts  in  vertical  direction;  then  the  component  of  p  parallel 
to  the  line  of  action  of  P  is 

dP  =  plcosOds  =^  cosddd,  (584) 

2i 

from  which 

P  =  pld, 

or  p  =  ^  (585) 

The  force  of  friction  on  the  elementary  strip  Ids  is  uplds,  and 
the  moment  of  this  force  about  the  axis  of  the  journal  is 


whence  by  integration 

M  =  ^  (586) 

The  work,  in  foot-pounds,  lost  per  minute  due  to  the  friction 
is  given  by  the  formula 

„,  fJ.TT2NPd  /ro^TN 

Wf  =       24     >  (587) 

in  which  the  diameter  d  is  expressed  in  inches,  and  N  denotes  the 
revolutions  per  minute. 

(6)  Uniform  vertical  wear. — The  statement  that  the  normal 
wear  is  proportional  to  the  work  of  friction  is  equivalent  to  say- 
ing that  the  normal  wear  n  is  equal  to  the  product  of  a  constant 


ART.  367]  FRICTION  LOSSES  IN  BEARINGS  537 

k,  normal  pressure  p,  and  the  diameter  dof  the  journal,  since  work 
is  proportional  to  the  product  of  p  and  d.     Hence 

n  =  kpd  (588) 

It  is  evident  that  the  normal  wear  of  a  journal  and  bearing 
is  greatest  at  the  bottom  and  becomes  zero  at  the  sides.  If 
the  journal  and  bearing  remain  cylindrical  after  being  worn  it 
is  apparent  that  the  vertical  wear  h  is  constant,  and  the  normal 
wear  at  any  point  of  the  surface  in  contact  will  be  given  by  the 
relation 

n  =  h  cose  (589) 

Combining  (588)  and  (589) 

p  =  C  cose,  (590) 

in  which  the  constant  C  =  T-V     Substituting   (590)   in   (584), 
we  get 

dP  =  —  cos2ede, 

from  which  the  total  load  upon  the  bearing  is 

P  =  ^  (591) 

The  moment  of  the  force  of  friction  about  the  axis  of  the  journal 
is 

dM  =  ^— —  cosede, 
whence 

M  =  ^^  (592) 

Eliminating  C  by  combining  (591)  and  (592),  we  get 

M  =  ^  Pd  (593) 

7T 

The  energy,  in  foot-pounds,  lost  per  minute  is  given  by  the 
expression 

Wf  =  — ~ —  (594) 

368.  Work  Lost  Due  to  the  Friction  on  a  Conical  Journal. — 

The  expressions  for  the  moment  of  friction  and  the  work  lost 


538 


FRICTION  LOSSES  IN  BEARINGS  [CHAP.  XIX 


due  to  the  friction  on  a  conical  journal,  having  the  dimensions 
shown  in  Fig.  307,  are  determined  in  a  manner  similar  to  that 
used  in  Art.  367. 

(a)  Pressure  uniformly  distributed.  —  On  the  assumption  of 
uniform  distribution  of  pressure,  the  vertical  component  of 
the  normal  pressure  on  an  elementary  area  is  given  by  the 
expression 


dP  = 


from  which 


FIG.  >07. 
The  force  of  friction  on  the  elementary  area  is  — >  and 


COS  a 
the  moment  of  this  force  about  the  axis  of  the  journal  is 

,2 


dM  =  - 


sin  a 


(597) 


whence 


sin 


r   -  r 


3  cos  a  [7*2  —  fj. 
The  energy,  in  foot-pounds,  lost  per  minute  is 

,.^2 


(598) 


(599) 


18  cos  a  \r\  -  r\_ 
(6)   Uniform  vertical  wear. — Assuming  that  the  vertical  wear 


ART.  368]  FRICTION  LOSSES  IN  BEARINGS  539 

h  of  the  journal  and  bearing  remains  constant  for  all  points;  the 
normal  wear  at  any  point  is 

n  =  h  cos  a  cos  0  (600) 

Since  the  normal  wear  is  proportional  to  the  work  of  friction, 
it  is  evident  that 

n  =  kpr  (601) 

Combining  (600)  and  (601),  we  get 


p  =  -^p  (602) 

Ji  cos  a 
in  which  the  constant  C  =  — T Substituting  the  value  of 

p  in  (595),  we  get 

C 

dP  =  -      -  cos26dddr: 
tan  a 

whence  the  total  load  P  upon  the  journal  becomes 

To  determine  an  expression  for  the  moment  of  friction,  substi- 
tute (602)  in  (597) ;  whence 

dM  =  —. r  cosddddr 

sin  a 

Integrating 


M  =  -        \rl-r\\  (604) 

sin  a  L  2 

Combining  (603)  and  (604)  in  order  to  eliminate  C,  the  magni- 
tude of  the  moment  of  friction  of  the  conical  journal  is  given  by 
the  expression 


M  =  ,  (605) 

T  COS  a 

in  which  d  denotes  the  mean  diameter  of  the  conical  journal. 

To  calculate  the  energy  lost  due  to  friction,  the  following 
formula  may  be  used  : 


TF,  -  (606) 

3  cos  a 

369.  Proportions  of  Journal  Bearings.  —  In  general  the  dimen- 
sions of  the  various  parts  of  a  bearing  are  determined  by  means 
of  empirical  formulas  which  are  based  upon  the  diameter  of  the 


540  BEARING  PROPORTIONS  [CHAP.  XIX 

shaft.     Such  formulas  usually  give  a  well-proportioned  bearing 
having  an  excess  of  strength. 

(a)  Common  split  bearings. — The  empirical  formulas  given 
below  are  based  upon  a  series  of  dimensions  obtained  from 
several  sizes  of  common  split  bearings  similar  to  the  type  repre- 
sented by  Fig.  292.  The  cap  is  held  down  by  either  two  or  four 
bolts,  studs  or  cap  screws,  the  number  depending  upon  the 
length  of  the  bearing.  In  the  following  formulas  the  symbols  d 
and  I  denote  respectively  the  diameter  of  the  shaft  and  the  length 
of  the  bearing : 


Outside  diameter  of  bearing  =  1.75  d  -f-  0.5" 

Span  of  bolts  =  1.7  d  +  0.7" 
Distance  between  bolts  =  0.5  I 

Size  of  bolts  =  %6  d  +  0.25" 
Thickness  of  babbitt  =  KG  d  +  0.125' 


(607) 


(b)  Pedestal  bearings. — The  pedestal  bearing  shown  in  Fig.  293 
is  provided  with  removable  bearing  shells  which  are  made  alike 
so  that  they  are  reversible.     The  shells  are  lined  with  babbitt 
metal  that  is  peened,  then  bored  and  scraped  to  exact  size.     This 
type  of  bearing  is  manufactured  by  the  Stephens-Adamson  Mfg. 
Co.  in  six  sizes  ranging  from  3l%6  to  9^  inches  in  diameter. 
The  empirical  formulas  given  below  were  derived  from  dimen- 
sions furnished  by  the  manufacturer,  and  the  various  symbols 
used  in  these  formulas  apply  to  the  key  drawing  of  Fig.  293. 

a  =  3.7  d  +  3"  m  =  1.47  d  +  0.25 

b  =  3.1  d  +  1.75"  n  =  1.5  d 

c  =  3d  +  0.8"  p  =  0.88  d  +  1.8" 

e  =  1.9  d  +  0.5"  q  =  0.7  d  +  0.75" 

/  =  g  +  1.5"  r  =  0.75  d  -  0.5" 

g  =  2  d  s  =  0.38  d  +  0.5" 

h  =  1.7  d  -  0.3"  t  =  0.5  d  +  0.1" 

k  =  1.43  d  +  1.3"  u  =  0.28  d  +  0.4" 

Diam.  of  bolts  for  base  =  0.14  d  -f  0.45"  ] 

Diam.  of  bolts  for  cap  =  0.2  d  -  0.08"  (609) 

Thickness  of  babbitt  =  0.025  d  +  0.18"  J 

(c)  Rigid  post  bearings. — In  Fig.  308  is  shown  a  form  of  bab- 
bitt-lined split  bearing  that  is  used  for  carrying  line-  or  counter- 
shafts when  it  is  necessary  to  support  the  latter  from  posts  and 


(608) 


ART.  369] 


BEARING  PROPORTIONS 

CO  CO 

i  CD  CO  CO  NTH  V-i 

V-i  VJ<  V*  VH  \00  \00  \CM  \IM  VH  \00  rH\  V*  WS 
CO\   rH\   rH\   K5\  CO\   CO\  rH\  rH\   Oi\  1O\  rH       C0\   rH 

CO  CO  CD 

-.  <0  CO  CO  \-H  VH  \TH 

!          \-rjl    V-,   \00    \r-l    \<M    \rH    \00   rH\  \HX   COX    \00    >C\ 

rH\   ld\  C0\   t>\  rH\   03\   10\   rH  CO\   rH       l>\   rH 


CO  CO  CD 

CO           NrH           \-i  \-H  CD              CO 

\N   \-H   \00   rH\  V*  C0\    \00  >f3\  V-i  \00   VH 

Ht\  «\  US\  IH       W\    r-l       t>\  i-l  rH     rH\  rH\    W\    rH\ 


CO  CO                                                                             CD 

CO                 VH  \rH                                              CD  CD                              \rH 

^i                 V-i  V*  COX  IO\  \00  \00  \00  \rH  \(N  VH  \00  \00  "2\ 

^                    O5\    CO\    i-H  rH       rH\    rH\    CO\    t>\  l-i\    O5\  IO\    t~\    rH 


CO       CO        CD 

VH  VH  VH 
i        IH\  C0\  irt\ 


\oo  \po  v-i  \<N  \CN  v-i  V*  \oo  io 

,H\    CO\    t-\    r-K    lH\    O5\    C0\    t~\    rH 


co     co          V-i  VH     co     co  CD 

VH  \rH  \cN  rH\  \T)<  \00  >O\  V-i  VH  V*  \GO  \<M  \rH 

rHrHt-Hr-lT-lT-HrH(M(M(M(N<M(N 


\(N   \C<)   \<M   \(M 

<M(M(N(M(M(MC^(M 


\00  \QO  XT)<  \CM  \rH  \QO 

>f5\    t>\  l-H\    r-i\    M\  rH\ 


\oo  \x  \T)< 

iH\   C0\   M\ 

CO   CO   CO   CO 


V-i     (M  ro      CO  CO  CO 

>0\  \M  \00  V1  \rH  \00  \-H  \00  \00  \rH 

rH        O!\   IQ\    rH\    IO\    »O\  IO\    U5\  CO\    O5\ 


rjH  \IM 

\   rH\ 


CO\   rH\   rH\  05\   rH\ 


\Tj< 
rH\ 


\QO  \00 

cr>  t^  oo  o> 


rH\  IO\  IC\  CO\ 


\00  \00  \00  \(M  V*  \00  \T}<  \00  \00  \00  \00  \rjl 

IO\    t-\    rH\    rH\    CO\    rH\    rH\    IO\    l>\    rH\    CO\    CO\ 

THT-HrHrHrHrHT-HOqiMC^CQ 


541 


542 


THRUST  BEARINGS 


[CHAP.  XIX 


columns.  Bearings  for  a  similar  service  but  furnished  with  oil 
rings  are  also  obtainable.  In  Table  98  are  given  the  dimensions 
pertaining  to  the  different  sizes  of  the  type  of  rigid  post  bearing 
illustrated  in  Fig.  308. 


THRUST  BEARING  CONSTRUCTION 

The  journal  bearings  discussed  in  the  preceding  articles  are 
not  suitable  for  supporting  vertical  shafts  carrying  heavy  rotating 


UK  |2     -I 


FIG.  308. 


parts  or  horizontal  shafts  subjected  to  heavy  pressures  acting  in 
a  direction  parallel  to  the  shaft.  However,  in  many  installations 
of  horizontal  shafts  subjected  to  axial  loads,  as  for  example, 
thrusts  due  to  bevel  and  worm  gearing,  the  ordinary  journal 
bearing  is  used  and  the  axial  pressures  are  taken  care  of  by  means 
of  one  or  more  loose  washers  located  between  the  supporting 
bearing  and  a  suitable  collar  on  the  shaft. 

In  all  thrust  bearings  the  speed  of  the  surfaces  in  contact  is 
a  maximum  at  the  outer  edge,  and  at  the  axis,  theoretically,  it  is 
zero.  At  any  point  of  contact  the  wear  is  proportional  to  the 
work  of  friction;  namely,  the  product  of  the  pressure  at  the  point 


ART.  371] 


COLLAR  THRUST  BEARINGS 


543 


and  the  velocity  of  the  point.  The  exact  distribution  of  the 
pressure  existing  between  the  contact  surfaces  is  not  known 
definitely,  but  very  likely  it  is  maximum  at  the  center,  and  for 
that  reason  a  well -designed  flat  pivot  bearing  should  have  ring 
contact.  This  form  of  contact  surface  is  produced  by  merely 
removing  some  of  the  metal  at  the  center. 

370.  Solid  Bearing  with  Thrust  Washers. — A  solid  bearing 
provided  with  thrust  washers  and  used  for  supporting  a  bevel- 
gear  transmission  is  shown  in  Fig.  309.  The  thrust  of  the  gear 
is  taken  up  by  a  single  bronze  washer,  while  that  of  the  pinion  is 
taken  up  by  three  washers,  the  two  outer  ones  being  made  of 
steel  and  the  other  of  bronze.  The  steel  washers  have  spherical 
faces  which  fit  into  the  spherical  seats  furnished  on  the  hub  of  the 


FIG.  309. 

pinion  and  the  end  of  the  bearing.  Frequently  bearings  of  this 
description  are  furnished  with  a  casing  for  enclosing  the  gear  and 
pinion  thus  permitting  the  gearing  and  washers  to  run  in  an  oil 
bath.  In  addition  to  providing  an  effective  means  of  lubrica- 
tion, the  casing  also  protects  the  workmen  from  coming  into  con- 
tact with  the  gearing.  The  thrust  due  to  worm  gearing  is  fre- 
quently taken  care  of  by  an  arrangement  similar  to  that  shown  in 
Fig.  309,  but  plain  washers  are  used  in  place  of  spherical  seated 
ones. 

371.  Collar  Thrust  Bearings. — (a)  Marine  thrust  bearings  — 
For  shafts  subjected  to  a  considerable  end  thrust,  as  for  example 
a  screw  propeller  shaft,  or  the  shaft  of  a  centrifugal  pump  or 
blower,  the  axial  load  is  generally  absorbed  by  a  special  type  of 


544 


COLLAR  THRUST  BEARINGS 


[CHAP.  XIX 


thrust  bearing,  commonly  called  a  collar  thrust  bearing.  Instead 
of  transmitting  the  axial  load  to  the  end  of  the  bearing,  the  shaft 
is  provided  with  a  series  of  collars  cut  integral  with  it,  which 
distribute  the  pressure  over  the  length  of  the  bearing. 

In  modern  marine  practice  the  rings  that  come  into  contact 
with  the  collars  on  the  shaft  are  made  in  the  shape  of  a  horseshoe. 
Such  a  construction  permits  their  removal  without  disturbing 
any  other  part  of  the  bearing.  The  lower  part  of  the  bearing 
housing  is  provided  with  a  reservoir  containing  oil,  and  in  order 
that  the  temperature  of  the  oil  may  not  become  excessive,  a  water 
coil  is  fitted  into  this  reservoir.  Each  end  of  the  housing  is 


FIG.  310. 

equipped  with  a  stuffing  box  so  that  the  level  of  the  oil  in  the 
reservoir  may  be  carried  slightly  above  the  lower  line  of  the  shaft, 
thus  insuring  ample  lubrication.  Each  of  the  bearing  rings  has 
an  independently  controlled  circulation  of  water,  thus  making  it 
possible  to  maintain  a  uniform  temperature  throughout  the 
bearing. 

(b)  De  Laval  thrust  bearing. — For  the  high  rotative  speeds  used 
on  certain  classes  of  centrifugal  pumps,  the  De  Laval  Steam  Tur- 
bine Co.  developed  a  babbitt-lined  ring-oiled  collar  thrust  bear- 
ing, the  details  of  which  are  clearly  shown  in  Fig.  310.  The 
collars,  instead  of  being  integral  with  the  shaft,  are  formed  on  a 
removable  steel  or  cast-iron  sleeve  which  is  fitted  to  the  impeller 
shaft  and  held  in  place  by  a  special  collar  and  lock  nut.  The 


ART.  371] 


COLLAR  THRUST  BEARINGS 


545 


babbitt-lined  bearing  shells  are  split  vertically,  while  the  pedestal 
or  housing  into  which  these  shells  are  fitted  is  split  horizontally. 
From  Fig.  310  it  is  evident  that  the  cap  of  the  bearing  is  not  sub- 
jected to  a  thrust,  but  the  entire  axial  load  comes  upon  the 
pedestal.  Vertically  split  shells  such  as  those  used  on  the  De 
Laval  bearing  are  easily  removed,  and  since  the  two  shells  are 
alike  they  may  be  interchanged.  The  oil  rings  are  made  of 
bronze  and  a  sufficient  number  are  provided  to  insure  ample 
lubrication. 

(c)  Bearing  for  combined  radial  and  axial  loads. — In  Fig.  311 
is  shown  a  design  of  a  combined  ring-oiling  journal  and  collar 


FIG.  311. 

thrust  bearing  that  is  used  on  a  single  suction  multistage  turbine 
pump.  The  bearing  is  babbitt-lined  throughout  and  ample 
lubrication  is  furnished  by  means  of  an  qil  ring.  The  housing  or 
bracket  a,  into  which  the  combined  bearing  shells  b  and  c  are 
fitted,  is  cored  out  so  that  water  may  be  circulated  through  it  in 
order  to  keep  the  bearing  cool.  The  tapped  hole  e  at  the  top  is 
connected  to  the  discharge  side  of  the  first  stage  while  the  hole/ 
at  the  bottom  is  connected  to  the  pump  suction.  The  shell  of  the 
bearing  is  split  horizontally.  To  provide  means  for  taking  up 
the  wear  of  the  thrust  collars,  an  adjusting  screw  g  and  lock  nut 
is  provided.  This  adjusting  screw  is  also  used  for  locating  the 
propeller  shaft  h  in  its  correct  position  relative  to  the  guide  vanes 
of  the  pump. 


546 


STEP  BEARINGS 


[CHAP.  XIX 


The  thrust  bearings  discussed  in  the  preceding  paragraphs  may 
be  located  at  any  convenient  point  along  the  shaft,  but  precau- 
tions should  be  taken  that  the  part  of  the  shaft  subjected  to  a 
thrust  will  not  be  too  long  or  it  may  tend  to  fail  by  a  buckling 
action  similar  to  a  long  column. 

372.  Step  Bearings.— (a)  Single-disc  type. — Frequently  a  form 
of  thrust  bearing  is  used  in  which  all  the  thrust  must  be  taken  up 
at  the  end  of  the  shaft,  as  for  example  a  vertical  transmission 


(a) 


FIG.  312. 


shaft.  For  slow  speeds  such  as  prevail  in  rotary  cranes  of  the 
jib  and  pillar  types,  the  thrust  due  to  the  load  and  weights  of 
moving  parts  are  usually  taken  care  of  by  an  ordinary  flat  pivot 
or  step  bearing  similar  to  the  designs  shown  in  Fig.  312  and  313. 
The  thrust  bearing  illustrated  in  Fig.  312 (a)  is  used  on  jib 
cranes  and  is  frequently  called  a  pintle  bearing.  The  pintle  or 
pin  is  subjected  to  a  radial  load  in  addition  to  the  axial  thrust. 
The  pin  in  the  design  represented  by  Fig.  312(6)  is  also  subjected 
to  a  combined  radial  and  axial  load  and  is  used  on  pillar  cranes. 


ART.  372] 


STEP  BEARINGS 


547 


(6)  Multiple-disc  type.  —  The  wear  upon  a  pivot  may  be  reduced 
materially  by  introducing  several  discs  between  the  end  of  the 
pin  or  shaft  and  the  housing  of  the  bearing.  Alternate  discs  are 
generally  made  of  bronze  or  brass  and  steel.  The  lower  disc 
should  be  fastened  to  the  bearing  proper,  and  the  upper  one 
should  be  fastened  to  the  shaft,  while  the  intermediate  ones  must 
be  free.  It  is  evident  that  each  disc  is  subjected  to  the  same  unit 
pressure,  hence  the  effect 
of  such  a  combination  of 
discs  is  to  reduce  the  wear, 
since  the  relative  velocity 
between  adjacent  discs  is 
decreased.  In  order  to 
lubricate  the  various  disc 
surfaces,  oil  is  introduced 
through  a  central  hole  and 
radial  grooves  cut  into  the 
faces  of  the  discs  serve  as 
distributors. 

An  application  of  the 
use  of  loose  thrust  discs  is 
shown  in  Fig.  313  which 
illustrates  a  special  step 
bearing  designed  by  the 
Pawlings  Harnischfeger  Co. 
and  used  for  supporting  a 
heavy  cantilever  jib  crane. 
In  bearings  of  this  kind 
the  base  casting  is  usually 
made  in  one  piece,  but  in 

this  case  it  is  made  in  two  parts,  the  base  a  and  the  cap  6, 
which  are  bolted  together  by  heavy  stud  bolts  and  special  cap 
screws.  The  bronze  bushed  bearing  shell  c  is  provided  with  two 
spherical  seats,  the  centers  of  which  are  located  at  the  center  of 
the  bearing,  as  shown  in  the  figure,  thus  insuring  proper  align- 
ment at  all  times.  The  thrust  due  to  the  load  upon  the  crane 
and  the  weight  of  the  crane  comes  upon  the  discs,  two  of  bronze 
and  one  of  steel,  and  is  transmitted  through  the  end  of  the  sheU 
c  to  the  spherical  seated  pivot  bearing  in  the  base  a.  The  hori- 
zontal pressure  due  to  the  load  and  weight  is  transmitted  to  the 
spherical  journal  bearing  in  the  base  a  and  cap  6.  The  method 


.  313, 


548 


FRICTION  OF  PIVOTS 


[CHAP.  XIX 


of  lubricating  the  loose  discs  is  clearly  shown  in  the  figure,  also 
the  method  used  for  fastening  the  bushing  e  in  the  shell  c. 

In  Fig.  314  is  shown  a  form  of  an  adjustable  step  bearing  that 
is  intended  for  use  at  the  base  of  a  vertical  shaft.  It  is  evident 
from  the  construction  that  such  a  bearing  cannot  take  care  of 
heavy  radial  loads.  Solid  or  split  bearings  must  be  provided 
for  the  radial  loads.  The  bearing  shell  is  babbitted  and  the 
axial  load  comes  upon  two  hardened  steel  discs  having  spherical 
faces  as  shown  in  the  figure.  The  housing  containing  the 
bearing  is  large  and  provides  ample  reservoir  capacity  for  the 
lubricant. 


FIG.  314. 

373.  Work  Lost  due  to  Pivot  Friction. — General  equations. — 
For  the  general  case  we  shall  assume  the  pivot  to  be  some  surface 
of  revolution,  as  shown  in  Fig.  3 15 (a),  the  equation  of  the  curve 
being  unknown.  Assume  any  point  C  at  a  distance  x  from  the 
axis  AB.  If  p  denotes  the  intensity  of  the  normal  pressure  at 
the  point  C,  the  total  pressure  on  an  annular  strip  of  width  ds 
and  radius  x  will  be  2  irxpds.  Since  the  normal  to  the  surface  at 
the  point  C  makes  an  angle  6  with  the  axis  AB,  the  vertical 
component  of  the  pressure  on  the  annular  strip  is 

dP  =  2  irpxcosdds  (610) 

If  ri  and  rz  denotes  respectively  the  smaller  and  larger  radii 
of  the  pivot,  the  integration  of  (610)  between  these  limits  will 
give  the  sum  of  the  vertical  components  and  this  sum  must  be 
equal  to  the  axial  load  or  thrust  P;  that  is 


P  =  2  Trfpxdx 


(611) 


The  value  of  the  integral  will  depend  upon  the  law  of  variation 
of  the  normal  pressure  p. 


ART.  373] 


FRICTION  OF  PIVOTS 


549 


The  force  of  friction  upon  an  annular  strip  of  width  ds  is 
2TTfjLpxds}  in  which  n  denotes  a  coefficient  of  friction;  hence  the 
moment  of  this  frictional  resistance  about  the  axis  of  rotation 
is 

dM  =  2  7Tfjipx2ds,  (612) 

from  which  we  obtain  the  following  general  expression  for  the 
moment  of  friction  of  a  pivot: 

(613) 


From  (613)  it  is  evident  that  the  value  of  M  depends  upon  the 
following  important  considerations: 


FIG.  315. 

1.  Upon  the  form  of  the  pivot,  that  is  upon  the  equation  of  the 
bounding  curve. 

2.  Upon  the  law  of  variation  of  the  normal  pressure  p. 

3.  Upon  the  law  of  variation  of  the  coefficient  of  friction  p. 
In  any  given  case  the  form  of  the  pivot  is  known,  but  the  laws 

of  variation  for  p  and  ju  are  not  known.  But  little  experimental 
work  has  been  carried  on  to  establish  such  laws.  A  common 
method  of  dealing  with  pivots  is  to  assume  that  the  coefficient  of 
friction  remains  constant  and  that  the  pressure  is  uniformly 
distributed.  The  assumption  of  uniform  pressure  distribution 
may  represent  fairly  well  the  condition  existing  when  the  pivot 
and  its  bearing  are  new,  but  would  seem  unwarranted  in  the 
case  of  a  pivot  that  has  been  worn  in.  A  more  reasonable  sup- 


550          FRICTION  OF  COLLAR  THRUST  BEARINGS     [CHAP.  XIX 

position  is  that  the  normal  wear  at  any  point  is  proportional  to 
the  work  of  friction. 

374.  Work  Lost  in  a  Collar  Thrust  Bearing.  —  (a)  Pressure 
uniformly  distributed.  —  With  the  assumption  that  the  normal 
pressure  p  is  the  same  at  all  points  of  the  surfaces  in  contact,  the 
magnitude  of  the  thrust  P  upon  a  collar  pivot  according  (611) 
is  given  by  the  following  expression: 

P  =  irp(r\  -  r\)  (614) 

From  (614)  it  is  apparent  that  the  uniformly  distributed  pressure 
p  is  equal  to  the  thrust  P  divided  by  the  area  of  the  collar. 

Substituting  in  (613)  the  value  of  p  obtained  from  (614)  and 
integrating,  assuming  ^  as  constant,  we  obtain  the  following 
expression  for  the  moment  of  friction  of  a  collar  bearing  : 


_  i  +  ry.  +  rft 

3    L       r2  +  ri 

The  work,  in  foot-pounds,  lost  per  minute  by  a  collar  pivot 
according  to  the  above  assumption  may  be  determined  by  the 
formula 


w 


-j- 


i 


in  which  the  dimensions  r2  and  ri  are  expressed  in  inches  and  N 
denotes  the  revolution  of  the  collar  per  minute. 

(b)  Uniform  vertical  wear.  —  Letting  n  denote  the  normal  wear 
of  the  collar,  the  statement  "the  normal  wear  at  any  point  is 
proportional  to  the  work  of  friction"  may  be  expressed  by  the 
relation 

n  =  kpx, 
from  which 

P  =  £  (617) 

Substituting  this  value  of  p  in  (611)  and  (613)  and  assuming  ju 
as  constant,  we  obtain  the  following  expression  forP  and  M: 


=£-  (ri  -  r,)  (618) 

r|  -  r?)  (619) 


ART.  376]  TOWER'S  EXPERIMENTS  551 

Eliminating  n  and  k  between  (618)  and  (619),  we  find  that 

M  =  ^  (r2  +  n),  (620) 

which  shows  that  the  moment  of  friction  of  a  collar  pivot  is  the 
same  as  that  of  a  ring  of  infinitesimal  breadth,  with  a  diameter 
equal  to  the  mean  diameter  of  the  pivot. 

Upon  the  foregoing  assumption,  the  work,  in  foot-pounds,  lost 
per  minute  by  a  collar  pivot  is 


W,  =  (r*  +  rO  (621) 

375.  Analysis  of  a  Flat  Pivot.  —  It  is  of  interest  to  consider 
briefly  the  theoretical  distribution  of  pressure  in  the  case  of  a 
pivot  in  which  the  surface  in  contact  is  not  a  ring  but  a  com- 
plete circle.  From  (617)  we  have 


n 


The  normal  wear  n  may  practically  be  assumed  as  constant,  hence 
the  pressure  p  at  any  point  varies  inversely  as  the  distance  of 
that  point  from  the  axis  of  the  pivot.  Theoretically  the  pressure 
at  the  axis  is  infinitely  great.  While  this  is  not  the  actual  state 
of  affairs,  there  is  doubtless  a  great  intensity  of  pressure  at  the 
axis  and  this  produces  a  crushing  of  the  material  as  experience 
with  flat  pivots  seems  to  show.  It  is  a  good  plan  therefore  to  cut 
out  the  material  at  the  center  of  the  pivot  as  shown  in  Fig.  315(6) 
thus  changing  its  surf  ace  of  contact  to  that  of  a  ring,  as  in  the  case 
of  a  collar  pivot.  The  curve  mn  in  Fig.  315(6)  is  an  equilateral 

n 

hyperbola  whose  equation  is  px  =  v,  and  it  also  shows  graphic- 
ally how  the  pressure  upon  the  contact  surfaces  varies. 

376.  Tower's  Experiments  on  Thrust  Bearings.—  (ft)  Collar 
bearings.  —  In  the  Proceedings  of  the  Institution  of  Mechanical 
Engineers,  1888,  p.  173,  Mr.  Beauchamp  Tower  reported  the 
results  of  a  series  of  experiments  on  a  collar  thrust  bearing  14 
inches  outside  diameter  and  12  inches  inside  diameter.  The 
surfaces  in  contact  consisted  of  a  mild-steel  ring  located  between 
two  rings  made  of  gun  metal.  Table  99,  giving  the  values  of  the 
coefficient  of  friction  for  the  various  speeds  listed,  was  compiled 
from  the  results  published  in  the  original  report.  The  coefficients 


552 


TOWER'S  EXPERIMENTS 


[CHAP.  XIX 


TABLE  99. — COEFFICIENTS  OF  FRICTION  FOB  COLLAR  THRUST  BEARINGS 

— TOWER 


Pressures 

Revolutions  per  minute 

Aver, 
values 

Total 

ib. 

50 

70 

90 

110 

130 

sq.  in. 

600 

14.7 

0.0450 

0.0646 

0.0433 

0.0537 

0.0642 

0.0541 

1,200 

29.4 

0.0375 

0.0481 

0.0496 

0.0489 

0.0475 

0.0463 

1,800 

44.1 

0.0357 

0.0399 

0.0361 

0.0357 

0.0371 

0.0369 

2,400 

58.8 

0.0286 

0.0375 

0.0361 

0.0373 

.0.0410 

0.0361 

2,700 

66.1 

0.0354 

0.0334 

0.0346 

0.0361 

0.0378 

0.0354 

3,000 

73.5 

0.0347 

0.0341 

0.0348 

0.0352 

0.0356 

0.0348 

3,300 

80.8 

0.0337 

0.0322 

0.0348 

Bearing 

0.0336 

3,600 

88.2 

0.0312 

0.0444 

seized 

0.0378 

of  collar  friction  as  determined  by  Tower  are  based  upon  the 
assumption  that  the  force  of  friction  was  concentrated  at  the  end 
of  the  mean  radius  of  the  collars.  From  the  results  given  in 
Table  99  it  is  apparent  that  the  coefficient  of  friction  is  practi- 
cally independent  of  the  speed  and  that  it  tends  to  decrease  as 
the  load  on  the  bearing  is  increased. 

(6)  Step  bearings. — In  Table  100  are  given  the  results  of  a  series 
of  experiments,  made  by  Tower,  on  a  flat  steel  pivot  3  inches  in 
diameter  running  on  a  manganese  bronze  step  bearing.  To  in- 
sure proper  lubrication  of  the  contact  surfaces,  the  oil  was  sup- 
plied to  the  center  of  the  pivot  and  distributed  by  a  single  dia- 
metrical groove  which  extended  to  within  J^e  inch  from  the 
circumference  of  the  pivot.  At  the  slower  speeds  the  oil  circula- 
tion, which  was  automatic  due  to  the  centrifugal  action,  was 

TABLE  100. — COEFFICIENTS  OF  FRICTION  FOR  STEP  BEARINGS — TOWER 


Revolutions  per  minute 


Ib.  per  sq.  in. 

50 

128 

194 

290 

353 

20 

0.0196 

0.0080 

0.0102 

0.0178 

0.0167 

40 

0.0147 

0.0054 

0.0061 

0.0107 

0.0096 

60 

0.0167 

0.0053 

0.0051 

0.0078 

0.0073 

80 

0.0181 

0.0063 

0.0045 

0.0064 

0.0063 

100 

0.0219 

0.0077 

0.0044 

0.0056 

0.0057 

120 

0.0221 

0.0083 

0.0052 

0.0048 

0.0053 

140 

.  \ 

0.0093 

0.0062 

0.0046 

0.0053 

160 

1 

0  0113 

0  0068 

0  0044 

0  .  0054 



ART.  377]  SCHIELE  PIVOT  553 

somewhat  restricted,  varying  from  20  to  56  drops  per  minute, 
while  at  the  higher  speed  the  bearing  was  flooded.  Due  to  the 
more  effective  lubrication  of  the  step  bearing  used  in  these  experi- 
ments, the  coefficients  of  friction  are  much  less  than  those  obtained 
with  the  experimental  collar  bearing.  The  coefficients  of  fric- 
tion as  given  in  Table  100  were  determined  from  the  moments  of 
friction  by  means  of  a  formula  based  on  the  assumption  that  the 
pressure  is  uniformly  distributed.  In  other  words  the  moment 
of  friction  is  M  =  %  /iPr. 

In  a  second  series  of  experiments,  a  white  metal  step  bearing 
was  used  in  place  of  the  manganese  bronze  bearing,  and  the  results 
obtained  gave  coefficients  of  friction  slightly  greater  than  those 
given  in  Table  100.  For  all  practical  purposes  the  coefficients 
given  in  Table  100  may  also  be  used  for  white  metal  bearings. 

377.  Schiele  Pivot.  —  It  is  possible  to  design  a  pivot  with  a  sur- 
face of  such  a  nature  that  the  pressure  between  the  pivot  and  its 
bearing  shall  be  the  same  at  all  points  of  contact.  From  Fig. 
31  5  (a),  it  is  evident  that  the  relation  between  the  normal  wear 
n  and  the  vertical  wear  h  is 

n  =  h  cosfl  (622) 

Combining  (617)  and  (622),  we  obtain  the  following  general  ex- 
pression for  the  normal  pressure: 

P  =  ^  (623) 

in  which  C  denotes  the  ratio  of  the  constants  h  and  k. 

Assuming  that  p  is  to  remain  constant  at  all  points  of  contact, 
it  follows  that  cos  6  is  proportional  to  x,  that  is 

cos  6  =  Kx  (624) 

Since  6  is  the  angle  that  the  normal  to  the  bounding  curve  of  the 
pivot  makes  with  the  axis  of  the  pivot,  we  have 

dy  =  t&nddx 
Differentiating  (624) 

Kdx  =  —  s'mddO, 
whence 


Integrating  (625),  we  get 

Ky  =  sin  0  -  loge  (sec  0  +  tan  0)  +  F 


554  SCHIELE  PIVOT  [CHAP.  XIX 

Eliminating  6  by  means  of  (624) 


Ky  =  ± 


This  is  the  equation  of  the  iradrix  or  sometimes  wrongly  called, 
the  antifriction  curve. 

T 

From  Fig.  3  15  (a),  cos0  =  -~~,  and  combining  this  with  (624), 

find 
therefore 


we  find  K  =        .  From  the  construction  of  the  tractrix  BC  = 


K  =  (627) 

Moment  of  friction.  —  The  moment  of  friction,  about  the  axis 
of  the  pivot,  of  the  normal  pressure  p  acting  on  an  annular  strip 
of  width  ds  is 

dM  = 


Assuming  as  in  the  preceding  discussions  that  the  coefficient  ju 
remains  constant,  we  obtain  by  integration  the  following  ex- 
pression for  the  moment  of  friction  : 


M  =  ju7rpr2  (r\  -  rl)  (628) 

For  uniform  distribution  of  pressure  it  was  shown  that 


Substituting  this  value  of  p  in  (628),  we  have 

M  =  »Pr2  (629) 


Comparing  (629)  with  the  expression  for  the  moment  of  fric- 
tion for  a  collar  pivot  as  given  by  (620),  it  is  evident  that  the 
moment  of  friction  of  a  Schiele  pivot  is  always  the  greater.  The 
Schiele  pivot  has  one  advantage  in  that  it  keeps  its  shape  as  it 
wears  and  it  is  self-adjusting.  Due  to  its  excessive  cost  of  manu- 
facture it  is  used  but  little. 

References 

Elements  of  Machine  Design,  by  W.  C.  UNWIN. 

Bearings  and  Their  Lubrication,  by  L.  P.  ALFORD. 

Friction  and  Lost  Work  in  Machinery  and  Mill  Work,  by  R.  H.  THURSTON. 

Lubrication  and  Lubricants,  by  L.  ARCHBUTT  and  L.  M.  DEELEY. 


ART.  377]  REFERENCES  555 

Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 

Theory  of  Lubrication,  Phil.  Trans.,  1886,  Part  I,  p.  157. 

Report  on  Experiments  on  Journal  Friction,  Proc.  Inst.  of  Mech.  Engrs., 
1883  and  1885. 

Report  on  Experiments  on  Collar  Friction,  Proc.  Inst.  of  Mech.  Engrs., 
1888,  p.  173 

Report  on  Experiments  on  Pivot  Friction,  Proc.  Inst.  of  Mech.  Engrs., 
1891,  p.  111. 

Bearings,  Trans.  A.  S.  M.  E.,  vol.  27,  p.  420 

Comparative  Test  of  Three  Types  of  Lineshaft  Bearings,  Trans.  A.  S.  M. 
E.,  vol.  35,  p.  593. 

On  the  Laws  of  Lubrication  of  Journal  Bearings,  Trans.,  A.  S.  M.  E.,  vol. 
37,  p.  167. 

Friction  Tests  of  Lubricating  Greases  and  Oils,  Bull.  No.  4,  Univ.  of  Mo., 
December,  1913. 

Bearings  for  High  Speed,  Traction  and  Transmission,  vol.  6,  No.  22. 

Experiments,  Formulas  and  Constants  of  Lubrication  of  Bearings,  Amer. 
Mach.,  vol.  26,  pp.  1281,  1316,  1350. 

Charts  for  Journal  Bearings,  Amer.  Mach.,  vol.  37,  p.  848. 

Charts  for  Journal  Bearings,  Amer.  Mach.,  vol.  39,  pp.  1017  and  1069. 

Lubrication  of  Bearings,  Amer.  Mach.,  vol.  45,  p.  847. 

Temperature  Tests  on  Journal  Bearings,  Power,  vol.  37,  p.  848. 

Bearing  Design  Constants,  Power,  vol.  43,  p.  251. 

Electrical  Machine  Bearings,  Power,  vol.  44,  p.  340 
,    Pressure  Oil-Film  Lubrication,  Power,  vol.  44,  p.  798. 

Experiments  with  an  Air-lubricated  Journal,  Jour.  A.  S.  Nav.  Engrs.,  vol. 
9,  No.  2. 

The  Kingsbury  Thrust  Bearing,  The  Electric  Journal. 

A  New  Type  of  Thrust  Bearing,  Trans.  Nat.  Elect.  Lt.  Assoc.,  1913. 


CHAPTER  XX 
BEARINGS  WITH  ROLLING  CONTACT 

378.  Requirements  of  Rolling  Contact. — A  bearing  having  a 
rolling  contact  is  one  in  which  the  journal  is  supported  on  rollers 
or  balls,  thereby  decreasing  the  frictional  resistance,  since  rolling 
friction  seldom  exceeds  sliding  friction  under  the  same  conditions 
of  load  and  operation.     Due  to  the  use  of  improved  machinery 
for  producing  the  rolling  elements,  bearings  with  rolling  contact 
are  now  used  for  all  classes  of  service.     A  bearing  of  this  kind  to 
be  commercially  successful  must  fulfill  the  following  conditions: 

(a)  The  arrangement  of  the  rolling  elements  should  be  such 
that  sliding  is  reduced  to  a  minimum. 

(6)  The  rolling  elements  must  all  be  of  the  same  size,  and  ac- 
curacy in  form  is  absolutely  essential. 

(c)  The  rolling  elements  must  be  extremely  hard  and  their 
surfaces  must  be  polished  very  smooth. 

(d)  The  rolling  elements  must  be  so  arranged  that  they  will 
not  run  off  their  guides  or  raceways. 

(e)  The  rolling  elements  must  not  be  overloaded,  as  they  may 
become  distorted  thus  changing  the  conditions  entirely. 

(/)  The  pressure  should  be  approximately  normal  to  the  sur- 
face of  contact. 

379.  Classification. — Bearings   with   rolling   contact   may   be 
divided,  according  to  the  kind  of  rolling  element  used,  into  the 
following  classes: 

(a)  Roller  bearings,  in  which  either  cylindrical  or  conical  rollers 
are  placed  between  the  journal  and  its  bearing. 

(6)  Ball  bearings,  in  which  hardened  steel  balls  are  used  in 
place  of  the  rollers. 

Each  of  the  above  classes  may  be  subdivided  into  the  following 
types:  (1)  radial  bearings',  (2)  thrust  bearings. 

ROLLER  BEARINGS 

380.  Radial  Bearing  having  Cylindrical  Rollers. — (a)  Moss- 
berg  bearing.^- The  simplest  form  of  roller  bearing  for  a  journal 

556 


ART.  380] 


RADIAL  ROLLER  BEARINGS 


557 


consists  of  a  sleeve  surrounded  by  a  series  of  cylindrical  rollers, 
rolling  inside  of  a  bored  casing  or  outer  race.  Fig.  316  shows  such 
a  bearing  made  by  the  Standard  Machinery  Co.  and  known  as 
the  Mossberg  bearing.  The  sleeves,  rollers,  and  outer  casings 
must  have  true  cylindrical  surfaces  and  'for  proper  working  the 
axes  must  always  remain  parallel  to  each  other.  To  keep  the 
rollers  d  in  the  desired  position,  they  are  placed  in  a  cage  c 
having  its  outside  diameter  slightly  less  than  the  inner  diameter 
of  the  outer  race  e,  and  its  internal  diameter  a  trifle  greater  than 
the  diameter  of  the  sleeve  6.  The  cage,  made  from  a  good  tough 
bronze  or  steel,  is  provided  with  a  series  of  slots  reamed  to  size, 


FIG.  316. 

into  which  the  steel  rollers  are  placed  as  shown  in  Fig.  316. 
No  doubt  there  is  a  certain  amount  of  sliding  between  the  roller 
and  the  cage,  but  actual  tests  seem  to  show  that  this  sliding 
action  reduces  the  efficiency  of  the  bearing  but  little. 

(b)  Norma  bearing. — A  form  of  roller  bearing  shown  in  Fig. 
317  has  been  recently  developed  and  placed  on  the  market.  It 
consists  of  an  outer  race  e  having  a  convex  or  ball-shaped  interior 
surface,  against  which  the  rollers  d  bear.  The  sleeve  or  race  6 
is  cylindrical  and  is  fastened  to  the  shaft  or  journal.  The 
cylindrical  rollers,  which  are  short,  are  held  in  alignment  by  the 
specially  constructed  steel  cage  c.  As  may  be  seen  in  Fig.  317, 
the  outer  race  e  is  open-sided  thus  facilitating  the  assembling, 
mounting  or  dismantling  of  the  bearing.  The  manufacturers  of 
this  bearing,  which  originated  in  Stuttgart,  Germany,  claim  that 


558 


RADIAL  ROLLER  BEARINGS 


[CHAP.  XX 


the  Norma  bearing,  as  it  is  called,  is  capable  of  supporting  greater 
loads  than  ball  bearings  having  the  same  dimensions  and  running 
at  the  same  speed.  These  bearings  are  made  so  that  they  are 
interchangeable  with  ball  bearings,  thus  providing  for  their 
application  in  cases  where  ball  bearings  fail 
under  the  applied  load. 

Due  to  inaccuracies  of  the  rolling  element  or 
to  wear,  the  rollers  in  an  ordinary  roller  bear- 
ing may  acquire  a  tendency  to  move  length- 
wise, thus  causing  more  or  less  end  pressure 
on  the  cage.  To  eliminate  this  end  pressure, 
holes  are  drilled  in  the  ends  of  the  rollers,  or  in 
the  cage,  and  steel  balls  are  inserted. 

381.  Radial  Bearings  having  Conical  Rollers. 

—The  rollers  instead  of  being  cylindrical  may 
be  conical  as  shown  in  the  bearing  illustrated  in 
Fig.  318.  In  its  general  construction  this  bear- 
ing is  similar  to  the  plain  roller  bearing.  It 
consists  of  a  series  of  conical  rollers  d  located  between  the  inner 
and  outer  cones  b  and  e.  The  cage,  consisting  of  two  rings  c  and  / 
made  of  high-carbon  steel,  is  provided  with  sockets  for  holding 
the  ends  of  the  rollers.  These  rings  are  held  together  by  stay  rods 
g,  shown  by  the  dotted  lines.  The  ends  of  the  rollers  are  beveled 


FIG.  318. 


FIG.  319. 


to  a  sligh  t  angle  and  bear  against  corresponding  shoulders  on  the 
cage  and  inner  cone  6.  To  insure  true  rolling  in  this  type  of 
bearing  it  is  necessary  that  all  the  axes  of  the  rollers  intersect 
the  axis  of  the  journal  in  a  common  point.  Bearings  having 
two  sets  of  conical  rollers  are  also  made  at  the  present  time. 


ART.  382] 


HYATT  BEARING 


559 


Timken  roller  bearing. — Another  successful  roller  bearing  using 
tapered  rollers  is  shown  in  Fig.  319.  It  is  used  rather  extensively 
in  automobile  construction  and  differs  from  the  bearing  just 
described  in  minor  details  only.  The  Timken  bearing  is  made  in 
various  styles  and  that  shown  in  Fig.  319  is  known  as  the  "short 
series." 

One  important  advantage  possessed  by  conical  roller  bearings 
over  any  other  form  of  roller  bearing  lies  in  the  provision  for 
taking  up  the  wear  if  there  is  any.  It  is  merely  necessary  to 
force  the  inner  cone  and  the  rollers  further  into  the  outer  cone 
or  cup. 

382.  Radial  Bearings  having  Flexible  Rollers. — Due  to  in- 
accuracy in  manufacturing,  slight  deflections  of  the  shaft,  yield- 


FIG.  320. 

ing  of  the  supports  or  mounting  of  a  roller  bearing,  a  roller  may 
move  out  of  its  correct  position  and  cause  the  line  of  contact 
with  the  sleeves  or  races  to  become  curved  instead  of  straight. 
Such  a  condition  would  cause  a  long  roller  of  brittle  material  to 
break  and  the  whole  bearing  would  thereby  be  ruined.  To  over- 
come this  difficulty  the  flexible  roller  has  been  devised  and  is 
now  used  for  all  classes  of  service.  A  form  of  bearing,  known  as 
the  Hyatt  bearing,  using  flexible  rollers  is  shown  in  Fig.  320. 
The  Hyatt  rollers  are  made  of  a  strip  of  steel  wound  into  a  coil 
or  spring  of  uniform  diameter.  Due  to  its  flexibility,  the  roller 
will  adjust  itself  to  any  irregularity  of  the  bearing  such  as  im- 
perfect alignment;  furthermore,  the  distribution  of  the  load  along 


560 


ROLLER  THRUST  BEARING 


[CHAP.  XX 


the  entire  length  of  the  roller  will  be  practically  uniform,  thus 
permitting  the  use  of  commercial  shafting,  hardened  and  ground 
journals  not  being  essential  except  under  extreme  conditions  of 
loading.  Due  to  its  construction  the  lubrication  of  the  Hyatt 
bearing  is  very  effective,  since  the  center  of  each  roller -is  really 
a  large  oil  reservoir.  The  Hyatt  bearing  is  also  made  with  a 
hardened  steel  inner  sleeve  which  may  be  fastened  to  a  soft 
steel  shaft. 

383.  Thrust   Bearings   having   Cylindrical   Rollers. — In   Fig. 
321  is  shown  a  thrust  bearing  using  cylindrical  rollers.     It  is 


FIG.  321. 

claimed  by  the  manufacturer  of  this  bearing,  that  while  theoretic- 
ally a  thrust  bearing  having  conical  rollers  is  better  than  one 
having  cylindrical  rollers,  the  theory  is  not  borne  out  in  actual 
operation.  The  explanation  no  doubt  lies  in  the  mechanical  in- 
accuracy of  the  various  parts  in  contact.  To  reduce  the  tendency 
of  the  cylindrical  rollers  to  groove  the  discs,  the  rollers  should 
travel  in  different  paths.  In  some  designs  this  is  accomplished  by 
placing  the  slots  in  the  cage  at  different  distances  from  the  shaft 
center.  Another  scheme  used  for  preventing  the  formation  of 
grooves  is  shown  in  Fig.  321,  and  consists  of  rollers  having  dif- 
ferent widths.  The  thrust  of  the  rollers  in  a  radial  direction 
may  be  taken  up  by  a  ball,  as  shown  in  the  figure. 


ART.  384] 


ROLLER  THRUST  BEARING 


561 


Fig.  321  shows  a  roller  thrust  bearing  installed  under  a  5,500- 
horse-power  turbine  generator  of  the  Niagara  Falls  Power  Co. 
The  maximum  load  coming  upon  this  bearing  is  190,000  pounds 
and  the  normal  load  is  156,000  pounds.  The  normal  speed  is 
250  revolutions  per  minute,  and  the  maximum  may  reach  500 
if  the  governor  fails.  By  consulting  Fig.  321  it  will  be  noticed 
that  on  the  under  side  of  each  cage  c  and  near  its  bore  is  located 
an  auxiliary  roller  bearing,  the  function  of  which  is  to  support 
the  weight  of  the  cage.  The  cages  of  both  the  main  and  auxiliary 
bearings  are  made  of  bronze,  while  the  thrust  discs  or  washers  are 
made  of  case-hardened  machinery  steel. 

384.  Thrust  Bearings   having   Conical  Rollers. — A  common 
form  of  thrust  bearing  using  conical  rollers  consists  of  a  cage  c,  a 
series  of  steel  rollers  d,  two  steel  thrust  discs  e  and  b,  and  an  exter- 
nal ring  /  as  shown  in  Fig.  322. 

The  cage  c,  generally  made  of 
one  solid  piece  of  metal,  is  pro- 
vided with  tapered  holes  into 
which  are  placed  the  conical 
rollers  d.  Due  to  the  action 
of  the  load  upon  the  bearing, 
the  rollers  tend  to  move  radi- 
ally outward,  and  to  reduce  FlG-  322« 
this  tendency,  the  apex  angle 

is  made  relatively  small.  One  prominent  manufacturer  makes 
this  angle  6  degrees.  The  thrust  discs  may  both  be  made  coni- 
cal, or  either  may  be  flat  and  the  other  coned;  in  other  words, 
the  axes  of  the  rollers  need  not  necessarily  be  at  right  angles  to 
the  axis  of  the  shaft.  However,  to  obtain  pure  rolling,  the  ver- 
tices of  the  rollers  and  of  the  conical  thrust  surfaces  must  be  in 
a  common  point  on  the  axis  of  the  shaft.  In  the  thrust  bear- 
ing shown  in  Fig.  322,  the  radial  thrust  of  the  rollers  is  taken 
care  of  by  the  tool-steel  ring  /.  In  some  designs  the  end  thrust 
of  the  rollers  against  the  cage  is  taken  by  a  ball  located  between 
two  cupped  surfaces. 

385.  Allowable  Bearing  Pressures  and  Coefficients  of  Friction. 

—The  intensity  of  pressure  coming  upon  the  elements  of  a  roller 
bearing  should  not  exceed  the  elastic  limit  of  the  material  or 
permanent  deformation  will  occur.  Such  deformation  ruins 
either  the  rollers  or  the  bearing  surfaces,  or  both. 


562  ROLLER  BEARING  DATA  [CHAP.  XX 

In  1898  Prof.  Stribeck,  the  well-known  head  of  the  Technical 
Laboratories  in  New  Babelsberg  carried  on  extensive  investiga- 
tions on  sliding,  roller  and  ball  bearings.  Among  the  tests  made 
was  a  series  investigating  the  relations  existing  between  the 
coefficient  of  friction,  specific  load,  and  the  speed  for  many  types 
and  sizes  of  bearings.  The  following  are  a  few  of  the  conclusions 
mentioned  in  a  report  submitted  by  Prof.  Stribeck. 

(a)  That  the  load  coming  upon  either  a  roller  or  ball  bearing 
may  be  considered  as  supported  by  one-fifth  the  number  of  rollers 
or  balls  in  the  bearing.     This  distribution  of  the  load  is  not  uni- 
form over  each  of  the  carrying  rollers  or  balls. 

(b)  That  the  ball  bearing  has  a  load  carrying  capacity  much 
in  excess  of  that  of  the  roller  bearing. 

(c)  That  the  roller  bearing  has  a  higher  coefficient,  of  friction 
than  the  ball  bearing  for  similar  conditions  of  speed  and  loading. 

(d)  That  the  coefficient  of  friction  for  ball  bearings  is  practic- 
ally a  constant  for  a  wide  range  of  speed  and  load. 

(e)  That  the  chief  advantage  of  roller  bearings  over  plain  bear- 
ings lies  in  the  lower  coefficient  of  friction. 

By  the  term  "specific  load"  is  meant  the  pressure  per  unit  of 
carrying  element.  For  a  plain  bearing  the  carrying  element  is 
considered  the  projected  area  of  the  journal.  For  roller  bearings 
the  carrying  element  is  considered  equivalent  to  one-fifth  of  the 
number  of  rollers  times  the  product  of  the  length  by  the  dia- 
meter of  the  rollers.  For  ball  bearings,  the  product  of  one-fifth 
of  the  number  of  balls  and  the  square  of  the  diameter  is  considered 
as  equivalent  to  the  carrying  element. 

386.  Roller  Bearing  Data. — Roller  bearings  for  motor  car  ser- 
vice have  been  standardized  to  such  an  extent  by  several  manu- 
facturers that  they  may  be  interchanged  with  ball  bearings  of 
similar  capacity. 

(a)  Norma  bearings. — In  Table  101  are  given  the  various  di- 
mensions of  the  medium  and  heavy-duty  Norma  roller  bearings. 
The  symbols  denoting  the  dimensions  refer  to  the  key  drawing  of 
Fig.  317.  The  load  capacity  as  given  in  this  table  is  based  on  a 
steady  load  and  slow  speed.  To  obtain  the  rating  at  any  partic- 
ular speed,  multiply  the  rating  given  in  Table  101  by  the  speed 
coefficient  obtained  from  Fig.  323.  The  chart  plotted  in  this 
figure  is  based  upon  data  deduced  from  the  load  ratings  given  in 
the  trade  publication  issued  by  The  Norma  Co.  of  America.  In 
addition  to  the  types  of  Norma  bearings  listed  in  the  table,  a 


ART.  386]  ROLLER  BEARING  DATA  563 

TABLE  101. — DATA  PERTAINING  TO  NORMA  ROLLER  BEARINGS 


Medium-duty  series 

Heavy-duty  series 

Dimensions 

Load 

Dimensions 

Load 

Size 

at 
10 

Size 

at 
10 

1 

2 

3 

4 

r.p.m. 

1 

2 

3 

4 

r.p.m. 

NM    25 

2.4410 

0.9842 

0.67 

0.04 

1,650 

NS    25 

3.1496 

1 

0.83 

0.08 

3,410 

NM    30 

2.8346 

1.1811 

0.75 

0.08 

2,150 

NS    30 

3.5433 

S 

0.91  0.08 

3,850 

NM    35 

3.1496 

1  .  3779 

0.830.08 

2,750 

NS    35 

3.9370 

"3 

0.980.08 

4,400 

NM    40 

3.5433 

1.57480.91 

0.08 

3,520 

NS    40 

4.3307 

T 

1.06 

0.08 

5,940 

NM    45 

3.9370 

1.7716 

0.98 

0.08 

4,400 

NS    45 

4.7244 

§ 

1.14 

0.08 

7,260 

NM    50 

4.3307 

1.9685 

1.06 

0.08 

5,280 

NS    50 

5.1181 

'3 

1.220.08 

8,580 

NM    55 

4.7244 

2.1653 

1.14 

0.08 

6,600 

NS    55 

5.5118 

§ 

1.30 

0.12 

9,460 

NM    60 

5.1181 

2.3622 

1.22 

0.08 

7,700 

NS    60 

5.9055 

g 

1.38 

0.12 

11,200 

NM    65 

5.5118 

2.5590 

1.30 

0.12 

8,360 

NS    65 

6.2992 

<N 

1.45 

0.12 

12,100 

NM    70 

5.9055 

2.7559 

1.38 

0.12 

10,120 

NS    70 

7.0866 

§ 

1.65 

0.12 

16,060 

NM    75 

6.2992 

2.9527 

1.46 

0.12 

11,880 

NS    75 

7.4803 

'1 

1.77 

0.12 

18,040 

NM    80 

6.6929 

3.1496 

1.54 

0.12 

12,980 

NS    80 

7.8740 

g 

1.88 

0.12 

18,260 

NM    85 

7.0866 

3.3464 

1.61 

0.12 

14,080 

NS    85 

8.4645 

2.00 

0.12 

19,140 

NM    90 

7.4803 

3  .  5433 

1.69 

0.12 

16,280 

NS    90 

8.8582 

w 

2.12 

0.12 

23,320 

NM    95 

7.8740 

3  .  7401 

1.77 

0.12 

17,380 

NS    95 

9.6456 

o> 

2.24 

0.12 

27,280 

NM  100 

8.4645  3.9370 

1.85 

0.12 

21,120 

NS  100 

10.4330 

1 

2.36 

0.12 

37,400 

^    0.8 


0.7 


"8     0.5 

8. 


0.4 


100 


200 


300 


400 


500 


1000 


1500 


2000 


Rev.    per    mm. 
FIG.  323. 


light-duty  bearing  is  also  manufactured;  furthermore,  each  of  the 
three  types  is  also  made  in  larger  sizes  than  those  given. 


564 


MOUNTING  OF  ROLLER  BEARINGS          [CHAP.  XX 


(b)  Hyatt  bearings. — The  dimensions  and  load-carying  capaci- 
ties for  the  long  and  short  series  of  the  Hyatt  high-duty  type  of 
roller  bearing,  similar  to  that  shown  in  Fig.  320,  are  given  in 
Table  102.  The  type  of  bearing  to  which  the  data  given  in  this 

TABLE  102. — DATA  PERTAINING  TO  HYATT  HIGH-DUTY  BEARINGS 


Short  series 


Long  series 


Size 

Dimensions 

Rating 

Size 

Dimensions 

Rating 

* 

2 

3 

1 

2 

3 

17,010 

.000 

2.249 

1.000 

460 

17,060 

2.000 

CO 

OJ 

co 

1,200 

17,012 

.000 

2.374 

1.125 

500 

17,062 

2.000 

<D 
CO 

0 

1,340 

17,014 

.125 

2.749 

1.250 

700 

17,064 

2.250 

-*J 

I 

1,700 

17,016 

.125 

2.875 

.375 

750 

17,066 

2.250 

O 

1 

1,900 

17,018 

.250 

3.375 

.500 

960 

17,068 

2.500 

OJ 

2,340 

17,020 

.250 

3.499 

.625 

1,040 

17,070 

2.500 

*» 

+-> 

2,530 

17,022 

.250 

3.625 

.750 

1,125 

17,072 

2.500 

£ 

42 

2,730 

17,024 

.250 

3.749 

.875 

1,200 

17,074 

2.500 

rt 

CO 
rt 

2,925 

17,026 

.375 

4.124 

2.000 

1,470 

17,076 

2.750 

0 

J 

3,490 

17,028 

.375 

4.249 

2.125 

1,550 

17,078 

2.750 

1 

<u 

1 

3,700 

17,030 

.375 

4.374 

2.250 

1,650 

17,080 

2.750 

a 

3,925 

17,032 

.500 

4.749 

2.500 

2,060 

17,082 

3.000 

-0 

'O 

4,820 

17,034 

1.500 

4.999 

2.750 

2,270 

17,084 

3.000 

ti 

S 

5,300 

17,036 

1.750 

5.374 

3.000 

3,030 

17,086 

3.500 

0> 

CD 

S 

6,890 

17,038 

1.750 

5.624 

3.250 

3,400 

17,088 

3.500 

A 

£ 

7,600 

table  applies  necessitates  the  use  of  a  heat-treated  or  hardened 
steel  shaft,  since  no  inner  shell  or  sleeve  is  furnished.  The  load 
ratings  specified  in  Table  102  represent  the  load  in  pounds  that 
any  particular  bearing  is  capable  of  carrying  at  a  speed  not  to 
exceed  1,000  revolutions  per  minute.  According  to  the  manu- 
facturer of  the  Hyatt  bearings,  the  load  capacity  at  1,500  revolu- 
tions per  minute  should  be  taken  as  equivalent  to  50  per  cent, 
of  that  given  in  the  table,  and  when  the  speed  is  500  revolutions 
per  minute  the  capacity  may  be  increased  50  per  cent,  above  that 
given  for  1,000  revolutions  per  minute. 

387.  Mounting  of  Roller  Bearings. — To  obtain  satisfactory 
service,  roller  bearings  of  all  types  must  be  carefully  protected 
from  water,  acids,  alkalies,  dust,  and  any  foreign  matter  that 
might  ruin  them.  Protection  may  be  obtained  by  housing  in  the 
bearing  and  sealing  the  openings  through  which  the  shaft  passes 
with  felt  packed  into  grooves  provided  for  that  purpose  in  the 


ART.  387] 


MOUNTING  OF  ROLLER  BEARINGS 


565 


end  or  cover  plates.  Filling  the  housing  and  bearing  with  a 
high-grade  stiff  grease,  provided  the  speed  is  not  too  high,  will 
also  aid  in  keeping  out  foreign  matter,  and  at  the  same  time  it 
will  furnish  the  necessary  lubrication. 

In  roller  bearings  of  the  Norma  and  Hyatt  type  both  the  inner 
and  outer  races  or  sleeves  are  rigidly  held  in  place.  Generally 
the  inner  race  is  made  a  light  driving  fit  on  the  shaft,  and  to  in- 
sure a  rigid  fastening,  the  race  should  be  clamped  between  a  suit- 
able shoulder  on  the  shaft  and  a  nut  provided  with  some  form  of 
locking  device.  For  various  forms  of  nut  locks  consult  Art.  80. 
The  outer  race  is  usually  clamped  between  a  shoulder  in  the  hous- 
ing and  an  outside  cover  plate,  or  in  some  cases  between  two 
cover  plates  The  shoulders  on  the  shafts  or  in  the  housing  against 


FIG.  324. 

which  a  bearing  is  clamped  should  be  sufficiently  high  to  provide 
ample  support  to  the  bearing,  if,  for  example,  the  shoulder  on 
the  shaft  be  made  too  small,  the  inner  race  when  pressed  into 
place  is  liable  to  sUp  over  this  shoulder  and  cause  the  race  to 
expand  slightly  thus  producing  undue  pressure  upon  the  end  of 
the  roller.  Since  Norma  and  Hyatt  bearings  cannot  take  an  end 
thrust,  the  latter  must  be  taken  care  of  by  suitable  thrust  bearings. 
In  mounting  a  conical  roller  bearing  either  the  cone  or  the  cup 
must  be  provided  with  means  for  taking  up  wear.  When  the 
inner  race  or  cup  is  mounted  on  a  non-rotating  member,  as  for 
example  on  the  front  wheel  spindle  of  a  motor  car,  it  is  considered 
good  practice  to  fasten  the  outer  race  or  cup  rigidly  into  the  hub 


566  FORMS  OF  RACEWAYS  [CHAP.  XX 

casting  or  forging,  and  provide  the  cone  with  an  adjustment  for 
taking  up  wear  by  making  it  an  easy  sliding  fit  on  the  spindle. 
When  the  cone  is  mounted  on  a  rotating  member,  good  practice 
dictates  that  the  cone  be  made  a  tight  press  fit  on  the  shaft  and 
that  the  wear  be  taken  up  by  making  the  cup  adjustable.  A  good 
example  of  an  installation  of  conical  roller  bearings,  in  which  the 
cones  are  mounted  on  rotating  members,  is  shown  in  the  rear- 
axle  worm-gear  transmission  of  Fig.  324.  Attention  is  directed 
to  the  fact  that  due  to  the  rigid  mounting  of  the  outer  races 
against  the  rim  of  the  end  cover  plates,  the  worm  shaft  is  always 
subjected  to  a  compression,  and  any  expansion  of  the  shaft  due 
to  an  excessive  rise  in  temperature  will  cause  it  to  deflect  a  small 
amount  and  necessarily  produce  undue  wear.  To  obviate  such 
a  condition,  the  bearings  may  be  so  arranged  that  the  cups  will 
come  against  shoulders  on  the  housing  or  gear  case,  thus  causing 
the  worm  shaft  to  be  in  tension. 

BALL  BEARINGS 

Formerly  ball  bearings  were  used  chiefly  for  light  loads,  but 
at  the  present  time  they  are  used  in  all  classes  of  machinery.  In 
general,  a  ball  bearing  consists  of  a  series  of  balls  held  by  a  suitable 
cage  between  properly  formed  hardened  steel  rings  called  races. 
These  races  may  be  of  such  shape  that  the  ball  has  two  points 
of  contact,  as  shown  in  Fig.  325,  or  it  may  have  three  or  even 
four  points  of  contact,  as  shown  in  Figs.  326  and  327,  respectively. 

388.  Forms  of  Raceways. — (a)  Two-point  contact. — The  sim- 
plest form  of  two-point  contact  is  the  flat  race  shown  in  Fig. 
325 (a).  In  this  construction  no  provision  is  made  for  retaining 
the  balls.  To  overcome  this  objection,  the  races  may  be  curved 
as  shown  in  Fig.  325(6),  (c)  and  (d),  the  latter  having  the  greatest 
carrying  capacity.  This  increase  of  carrying  capacity  is  no 
doubt  due  to  the  increased  area  of  contact.  For  a  well-designed 
ball  bearing  the  wear  upon  the  inner  and  outer  races  should  be 
the  same,  which  means  that  the  contact  pressure  upon  these 
races  should  be  the  same.  The  contact  pressure  depends  upon 
the  small  contact  area,  and  if  these  areas  are  to  be  equal  it  is 
necessary  that  the  radius  of  curvature  of  the  outer  race  should 
be  increased.  This  has  been  done  in  the  bearing  shown  in  Fig. 
325  (d). 


ART.  388] 


FORMS  OF  RACEWAYS 


567 


(6)  Three-point  contact. — In  Figs.  326  and  327  are  shown  forms 
of  bearing  raceways  having  three  and  four  points  of  contacts, 
respectively.  To  produce  true  rolling  of  the  ball  the  races  must 


FIG.  325. 


(a) 


(b) 


FIG.  326. 


be  laid  out  correctly.  Referring  to  Fig.  326(6),  and  letting  A, 
B  and  C  represent  the  three  points  of  contact,  extend  the  line  AB 
until  it  intersects  the  center  of  the  shaft  at  0,  also  draw  OF 
through  the  center  of  the  ball.  This  latter  line  represents  the 


568 


CONCLUSIONS  OF  STRIBECK 


[CHAP.  XX 


axis  of  rotation  of  the  ball,  and  the  lines  AE  and  BD  are  projec- 
tions of  two  circles  of  rotation.  From  similar  triangles  we  have 
that  AE  and  BD  are  proportional  to  OA  and  OB  respectively; 
therefore  it  is  evident  that  there  is  no  slipping  at  the  points  A 
and  B  and  that  the  desired  true  rolling  is  obtained.  The  third 
point  of  contact  C  is  determined  by  drawing  OC  tangent  to  the 
ball.  To  avoid  excessive  wedging  of  the  ball,  the  angle  a  should 
be  made  not  less  than  30  degrees. 

(c)  Four-point  contact. — In  either  of  the  four-point  bearings 
shown  in  Fig.  327,  the  pure  rolling  of  the  ball  is  obtained  when 


(a) 


FIG.  327. 


AD 
BC 


OA 
OB' 


The  various  lines  required  for  laying  out  a  bearing 


of  this  kind  are  drawn  in  a  general  way,  according  to  the  method 
outlined  for  the  three-point  bearing  in  the  preceding  paragraph. 

389.  Experimental  Conclusions  of  Stribeck. — Prof.  Stribeck 
in  his  investigation  of  bearings  having  rolling  contact  determined 
how  the  carrying  capacity  of  a  ball  was  affected  by  the  form  of  the 
raceway.  For  this  purpose  ball  bearings  having  raceways  shown 
in  Figs.  325  to  327  inclusive,  except  the  form  shown  in  Fig.  325  (d), 
were  used. 

Some  of  the  conclusions  arrived  at  were  as  follows: 

(a)  The  form  of  raceway  shown  in  Fig.  325  (a)  had  the  least 
frictional  resistance. 

(b)  An  increase  in  the  number  of  points  of  contact,  as  shown  in 
Figs.  326  and  327,  resulted  in  higher  frictional  resistances.     It  is 


ART.  390]  RADIAL  BALL  BEARINGS  569 

probable  that  due  to  imperfect  workmanship  the  conditions  re- 
quired for  pure  rolling  were  not  met. 

(c)  The  carrying  capacities  for  the  forms  of  raceways  shown 
in  Figs.  326  and  327  were  practically  the  same.     Theoretically 
the  four-point  contact  should  carry  more,  but  due  to  difficulties 
in  constructing  and  adjusting  such  a  bearing  it  is  almost  impossi- 
ble to  distribute  the  load  uniformly  over  the  various  points  of 
contact. 

(d)  The  carrying  capacity  for  the  form  of  raceways  shown  in 
Fig.  325 (c)  is  considerably  greater  than  for  the  other  forms  shown. 

(e)  The  frictional  resistance  for  the  form  indicated  by  Fig. 
325  (c)  is  a  trifle  greater  than  for  the  others,  but  practically  it 
may  be  considered  the  same. 

390.  Radial  Ball  Bearings. — (a)  Single-row  bearings. — A  radial 
ball  bearing  is  used  for  supporting  loads  acting  at  right  angles  to 
the  axis  of  rotation.  At  the  present  time  the  two-point  contact 
type  having  circular  raceways  is  used  almost  exclusively.  Such 
a  bearing  consists  of  an  outer  and  inner  race,  both  provided  with 
curved  ball  raceways  that  are  uniform  and  unbroken  around  the 
entire  circumference.  Between  these  races  is  located  a  series  of 
balls  separated  either  by  an  elastic  separator  or  by  a  bronze  or 
alloy  cage,  as  shown  in  Fig.  325.  The  type  of  elastic  separator 
mentioned  consists  of  a  short  helical  spring  fitted  with  suitable 
bearing  plates.  This  separator  was  formerly  used  in  the  Hess- 
Bright  bearings  and  at  the  present  time  is  still  used  under  certain 
special  conditions.  The  majority  of  the  separators  or  cages  now 
in  use  are  made  of  brass  or  bronze  and  steel  and  their  construc- 
tion makes  them  more  or  less  elastic. 

(6)  S.  K.  F.  bearing. — Radial  bearings  having  two  or  more 
rows  of  balls,  examples  of  which  are  shown  in  Figs.  325  (d)  and 
328,  have  also  been  devised.  In  selecting  this  type  of  bearing 
it  must  not  be  assumed  that  doubling  the  number  of  balls  neces- 
sarily doubles  the  load  capacity,  for  the  accuracy  of  workmanship 
required  for  such  a  condition  is  not  always  feasible. 

The  bearing  shown  in  Fig.  325  (d)  originated  in  Sweden  and  is 
known  as  the  S.  K.  F.  bearing.  The  outer  ball  race  e  is  a  ma- 
chined and  ground  spherical  surface,  the  center  of  which  lies  on 
the  axis  of  the  bearing.  The  inner  race  b  has  two  curved  ball 
raceways  having  a  radius  slightly  larger  than  the  radius  of  the 
ball.  The  balls  are  staggered  and  are  retained  by  the  phosphor 
bronze  separator  or  cage  c.  This  type  of  bearing  may  be  dis- 


570 


THRUST  BALL  BEARINGS 


[CHAP.  XX 


mantled  very  readily  by  swinging  the  race  b  and  the  balls  together 
within  the  outer  race  e,  and  then  removing  two  adjacent  balls  on 
either  side  diametrically  opposite  to  each  other.  This  operation 
permits  the  withdrawal  of  the  complete  center  portion  of  the  bear- 
ing. Another  important  advantage  of  the  S  K,  F.  bearing  is  its 
self-aligning  feature,  which  compensates  for  shaft  flexure  or 
deformation. 

(c)  Norma  bearing. — In  Fig.  328  is  shown  a  form  of  double-row 
ball  bearing  manufactured  by  The  Norma  Co.  of  America.  It 

consists   of   two   outer    races 

4..^  mounted   side   by  side  on  a 

single  inner  race  provided  with 
two  raceways.  The  raceways 
in  the  outer  rings  are  ground 
to  the  same  radius  as  that  used 
on  the  inner  race,  but  one-half 
of  the  shoulder  is  ground 
away  to  form  a  cylindrical 
surface,  tangent  to  the  circu- 
lar raceway,  as  shown  in  the 
figure.  It  is  evident  that  this 
form  of  outer  race  differs 
materially  from  that  shown  in 
Fig.  325 (c).  The  main  ad- 
vantage of  the  construction 
used  on  the  Norma  bearing 
lies  in  the  ease  with  which 
that  bearing  can  be  assembled 
and  dismantled  for  inspection. 
The  separator  used  consists 
of  alight  one-piece  bronze  ring 

having  a  channel  section.  The  flanges  of  this  ring  separator  are 
provided  with  spherical  seats  between  which  the  balls  are  held 
with  a  slight  elastic  pressure;  thus  the  balls  and  separator  may 
be  removed  as  a  single  unit. 

391.  Thrust  Ball  Bearings. — (a)  Two-point  type. — The  modern 
ball  thrust  bearing  is  made  with  either  two-  or  four-point  contact. 
In  Fig.  325  (a)  is  shown  a  two-point  contact  having  flat  raceways. 
It  consists  of  two  hardened  steel  plates  or  thrust  discs  b  and  c  be- 
tween which  is  located  the  cage  c  containing  the  balls.  The 
cage  may  be  made  of  one  piece  by  drilling  the  holes  for  the  balls 


FIG.  328. 


ART.  391] 


THRUST  BALL  BEARINGS 


571 


almost  through,  then  inserting  the  ball  and  by  means  of  a  special 
setting  tool  closing  in  the  upper  edge.  The  cage  may  also  be 
made  in  two  pieces  as  shown  in  the  figure. 


FIG.  329. 

The  type  of  thrust  bearing  having  curved  or  grooved  raceways 
is  shown  in  Figs.  329  and  330(a).  The  constructive  features  are 
clearly  shown  in  the  figures.  These  bearings  are  known  as  the 


(b) 


FIG.  330. 


full  ball  or  without  separator  type,  and  are  intended  for  very 
heavy  service  at  a  slow  speed.  The  type  shown  in  Fig.  330 (a) 
being  made  in  small  sizes  is  intended  for  use  on  automobile  steer- 
ing pivots,  while  that  illustrated  in  Fig.  329  is  made  in  the  larger 


572 


THRUST  BALL  BEARINGS 


[CHAP.  XX 


sizes  and  is  used  on  crane  hooks.  Another  type  of  thrust  bearing 
having  curved  raceways  is  shown  in  Fig.  330(6).  The  balls  are 
separated  by  a  cage  made  of  brass  or  special  alloy. 

All  thrust  bearings  thus  far  shown  are  intended  to  take  the 
thrust  in  one  direction  only.  In  cases  where  the  thrust  has  to 
be  taken  care  of  in  both  directions,  a  form  known  as  the  double- 
thrust  bearing  is  used.  Such  a  bearing,  shown  in  Fig.  331, 
consists  of  a  central  grooved  disc  /  securely  fastened  to  the  shaft, 
two  thrust  discs  b  and  e  having  grooves  to  correspond  with  those 
on  /,  and  two  phosphor-bronze  cages  c  retaining  the  balls.  The 


FIG.  331. 

form  of  bearing  just  described  may  be  so  arranged  that  the  com- 
bination of  balls,  cages,  and  thrust  discs  form  a  part  of  a  sphere 
as  shown  in  Fig.  331.  The  entire  combination  is  then  free  to 
revolve  in  a  specially  constructed  hardened  steel  casing  g.  To 
permit  easy  assembling  two  recesses  are  located  in  a  convenient 
position  on  one  side  of  the  casing  g. 

(b)  Four-point  type. — A  four-point  bearing  made  by  the  Auburn 
Ball  Bearing  Co.  is  shown  in  Fig.  332.     All  thrust  bearings  made 
by  this  company  are  of  the  four-point  type  and  have  no  separator 
for  the  balls.     The  condition  for  pure  rolling  is  fulfilled  as  may  be 
seen  from  the  geometry  of  the  figure. 

(c)  Leveling  washer. — In  any  thrust  bearing  it  is  always  desir- 
able to  distribute  the  load  uniformly  over  the  entire  series  of 


ART.  392] 


RADIO-THRUST  BEARINGS 


573 


balls.  This  is  done  by  providing  one  of  the  thrust  discs  with  a 
spherical  surface  thus  permitting  it  to  adjust  itself.  The  con- 
struction is  shown  in  Figs.  330  and  331.  When  both  discs  are 
flat  as  is  the  case  of  Figs.  329  and  332,  a  special  leveling  washer 
having  a  spherical  seat  should  be  used  in  connection  with  the 
stationary  thrust  disc. 

392.  Combined  Radial  and  Thrust  Bearing. — A  combination 
radial  and  thrust  bearing  is  used  in  places  where  provision  must 
be  made  for  both  radial  and  axial  loads.  Some  of  these  bearings 
are  so  arranged  that,  in  addition  to  the  radial  loads,  they  will  take 
care  of  a  thrust  in  only  one  direction  or  in  both  directions. 


FIG.  332. 

(a)  Radax  bearing. — A  bearing  known  as  Radax,  manufactured 
by  the  New  Departure  Mfg.  Co.,  is  used  for  both  radial  and  one 
direction  axial  loads.  The  details  of  this  bearing  are  clearly 
shown  in  Fig.  325(6) .  The  bearing  differs  from  an  ordinary  radial 
bearing  in  that  the  ball  raceway  has  a  two-point  angular  contact 
instead  of  radial  contact.  The  Radax  bearing  may  readily  be 
assembled  and  dismantled  since  the  inner  race,  separator,  and 
balls  may  easily  be  withdrawn  from  the  outer  race.  These  bear- 
ings are  made  interchangeable  with  corresponding  sizes  of  stand- 
ard radial  bearings. 

(6)  Gurney  radio-thrust  bearing. — In  Fig.  333  are  shown  the 
details  of  a  combined  radial  and  thrust  bearing  manufactured  by 
the  Gurney  Ball  Bearing  Co.  The  points  of  contact  between 
the  balls  and  the  inner  and  outer  races  are  not  on  radial  lines, 
but  lie  on  the  lines  that  intersect  the  axis  of  the  bearing  at  the 
point  0,  as  shown  in  the  figure.  The  steel  separator  is  made  in  a 
single  piece  having  a  light  but  rigid  construction.  The  radio- 
thrust  bearing  is  well  adapted  to  installations  in  which  there  is 
a  combination  of  radial  and  axial  loads,  and  where  the  latter  ex- 


574 


RADIO-THRUST  BEARINGS 


[CHAP.  XX 


ceeds  approximately  25  per  cent,  of  the  former.  It  is  generally 
conceded  that  ordinary  radial  bearings  should  not  be  subjected 
to  an  axial  load  exceeding  25  per  cent,  of  the  radial  load.  With 
relation  to  thrust  capacity,  the  Gurney  radio-thrust  bearings 
are  made  in  three  types.  Each  of  these  types  is  made  in  three 
series,  namely,  the  light,  medium,  and  heavy,  and  as  far  as  the 
dimensions  are  concerned  these  bearings  are  interchangeable 
with  the  corresponding  sizes  of  standard  radial  bearings. 

(c)  Double-row  and  duplex  bearings. — The  bearings  discussed 
in  the  preceding  paragraphs  are  used  in  places  where  the  axial 


FIG.  333. 


FIG.  334. 


loads  are  always  in  the  same  direction.  There  are,  however, 
many  places  requiring  bearings  capable  of  taking  a  thrust  in 
either  direction.  Such  a  condition  can  be  successfully  met  by 
installing  Duplex  bearings  which  consist  of  two  radio-thrust  bear- 
ings mounted  side  by  side,  or  by  using  a  double-row  combined 
radial  thrust  bearing.  A  bearing  of  the  latter  type,  made  by  the 
New  Departure  Mfg.  Co.,  is  shown  in  Fig.  334.  It  consists  of  a 
single  inner  race  containing  two  raceways,  a  bronze  separator 
made  in  two  pieces,  two  rows  of  balls  having  two-point  angular 
contact,  two  outer  races,  and  a  thin  steel  shell  which  is  closed  in 
over  the  outer  races,  as  shown  in  the  figure,  after  the  bearing  is 
assembled. 


ART.  393] 


BEARING  PRESSURES 


575 


393.  Allowable  Bearing  Pressures.  —  (a)  Load  per  ball  —  Ac- 
cording to  Stribeck,  the  carrying  capacity  in  pounds  per  ball  may 
be  determined  by  the  formula 

w  =  kd2  (630) 

in  which  d  denotes  the  diameter  of  the  ball  in  inches,  and  k 
a  constant  depending  upon  the  form  and  material  of  the  raceway 
and  the  speed  of  the  bearing  in  which  the  ball  is  used.  The 
following  values  of  k,  due  to  Stribeck,  are  based  upon  a  large 
number  of  experiments  on  bearings  in  which  the  races  were  made 
of  a  good  quality  of  hardened  steel  : 

1.  For  a  flat  or  conical  raceway  having  three  or  four  points  of 
contact  the  value  of  k  varies  from  420  to  700. 

2.  For  curved  raceways  whose  radius  of  curvature  equals  %  d 
and  having  two-point  contact  the  value  of  k  is  1,400. 

3.  For  special  races  and  balls  made  of  special  alloy  steel  the 
above  values  may  be  increased  50  per  cent. 

(6)  Load  per  bearing.  —  According  to  Art.  385(a),  the  total  load 
upon  a  ball  bearing  may  be  assumed  as  being  supported  by  one- 
fifth  of  the  number  of  baUs  in  the  bearing,  hence  multiplying 


the  values  of  w  by  —  we  get  the  total  load 
5 


W  = 


(631) 


in  which  Z  denotes  the  number  of  balls  in  the  bearing. 

(c)  Crushing  strength  of  balls.  —  In  Table  103  are  given  the 
approximate  crushing  strengths  of  the  commercial  sizes  of 
regular  tool  steel  balls.  These  values  according  to  R.  H.  Grant 
are  considered  reliable,  and  were  adopted  by  the  manufacturers 

TABLE  103.  —  CRUSHING  STRENGTH  OF  TOOL-STEEL  BALLS 


Diam.  of 
ball 

Ultimate 
strength,  Ib. 

Diam.  of 
ball 

Ultimate 
strength,  Ib. 

Diam.  of 
ball 

Ultimate 
strength,  Ib  . 

He' 

390 

H 

14,000 

lMe 

88,000 

%2 

875 

Ke 

19,100 

1 

100,000 

H* 

1,562 

1A 

25,000 

IK 

125,000 

X 

2,450 

Ms 

31,500 

IK 

156,000 

3/l6 

3,496 

5/8 

39,000 

IK 

225,000 

7A2 

4,780 

*4 

56,250 

IH 

263,000 

1A 

6,215 

13/16 

66,000 

IH 

306,000 

Me 

9,940 

7/8 

76,000 

2 

400,000 

576        .  BALL  BEARING  DATA  [CHAP.  XX 

after  several  years  of  testing.  Data  pertaining  to  special  alloy 
steel  balls  are  not  available,  but  it  is  safe  to  assume  that  the 
crushing  loads  will  exceed  those  given  in  Table  103  by  25  to 
50  per  cent.  According  to  Grant  a  factor  of  safety  of  ten  should 
be  used  in  selecting  balls  for  bearings. 

394.  Coefficient  of  Friction. — In  order  to  compare  ball  bearings 
with  ordinary  plain  bearings,  the  coefficient  of  friction  is  referred 
to  the  diameter  of  the  shaft.     Stribeck  found  experimentally 
that  the  coefficient  of  friction  of  a  good  radial  ball  bearing  having 
curved  raceways  is  independent  of  the  speed  within  wide  limits 
and  has  an  average  value  of  0.0015.     This  coefficient  will  prac- 
tically be  double  this  value  when  the  load  on  the  bearing  is  re- 
duced to  approximately  one-tenth  of  the  maximum  load.     The 
magnitude  of  the  coefficient  of  friction  in  a  radial  bearing  will 
also  depend  upon  the  axial  thrust  coming  upon  it.     According 
to  some  experimental  data  published  in  the  American  Machinist 
of  March,  1909,  the  coefficient  of  friction  for  a  radial  ball  bearing 
subjected  to  a  constant  radial  load  and  a  variable  axial  thrust 
increased  from  0.004  at  a  speed  of  200  revolutions  per  minute 
to  0.012  at  a  speed  of  1,200  revolutions  per  minute. 

395.  Ball  Bearing  Data.— Through  the  efforts  of  the  Com- 
mittee on  Standards  appointed  by  the  Society  of  Automobile 
Engineers,  practically  all  types  of  radial  ball  bearings  have  been 
standardized.     In  a  report  submitted  to  the  society  at  the  Spring 
meeting  in  1911  were  included  tables  giving  standard  dimensions 
of  light,  medium,  and  heavy  radial  bearings.    Some  manufacturers 
make  a  fourth  series  known  as  the  extra  heavy.    According  to  the 
catalogs  of  the  various  prominent  manufacturers  thrust  bearings 
are  made  in  light,  medium,  and  heavy  series.     With  few  excep- 
tions the  ball  bearing  manufacturers  have  adopted  the  English 
unit  for  the  ball  dimensions  and  the  metric  unit  for  the  remaining 
dimensions  of  the  bearing. 

(a)  Hess-Bright  radial  bearings. — In  Table  104  are  given  the 
leading  dimensions  of  the  light,  medium,  and  heavy  series  of  the 
wide-type  Hess-Bright  radial  bearings.  The  symbols  denoting 
the  dimensions  refer  to  the  key  drawing  of  Fig.  325  (c).  The  load- 
carrying  capacity  as  given  in  Table  104  is  based  on  a  steady  load 
and  a  constant  speed  not  exceeding  200  revolutions  per  minute. 
The  load  rating  of  any  size  bearing  operating  at  any  given  speed 
not  exceeding  1,500  revolutions  per  minute  may  be  determined 


ART.  395]  HESS-BRIGHT  RADIAL  BEARINGS  577 

TABLE  104. — DATA  PERTAINING  TO  HESS-BRIGHT  RADIAL  BEARINGS 


No.  and  type 

Dimensions  in  mm. 

Diam.  of 

Capacity 

of 
bearing 

1 

2 

3 

of 
balls 

at 
200  r.p.m. 

200 

30 

10 

9 

Me 

130 

201 

32 

12 

10 

M  6 

145 

202 

35 

15 

11 

Me 

165 

203 

40 

17 

12 

Ha 

240 

204 

47 

20 

14 

350 

205 
206 

52 
62 

25 
30 

15 
16 

if. 

395 
550 

207 

72 

35 

17 

Me 

660 

1 

208 

80 

40 

18 

860 

209 

85 

45 

19 

H 

945 

o 

210 

90 

50 

20 

H 

990 

»J 

211 

100 

55 

21 

Ke 

1,255 

1 

212 

110 

60 

22 

i  / 

1,630 

213 

120 

65 

23 

M 

1,760 

214 

125 

70 

24 

x^ 

1,870 

215 

130 

75 

25 

5K  e 

2,200 

216 

140 

80 

26 

% 

2,750 

217 

150 

85 

28 

1Me 

3,080 

218 

160 

90 

30 

H 

3,630 

219 

170 

95 

32 

3/ 

/4 

3,850 

220 

180 

100 

34 

13/i6 

4,180 

221 

190 

105 

36 

H 

4,840 

222 

200 

110 

38 

H 

5,280 

300 

35 

10 

11 

1A 

220 

301 

37 

12 

12 

H 

265 

302 

42 

15 

13 

X4 

285 

303 

47 

17 

14 

Me 

395 

304 

52 

20 

15 

Me 

440 

305 

62 

25 

17 

H 

660 

306 

72 

30 

19 

He 

880 

CQ 

307 

80 

35 

21 

1,100 

308 

90 

40 

23 

Me 

1,430 

&> 

309 

100 

45 

25 

% 

1,760 

03 

310 

110 

50 

27 

1Me 

2,090 

£ 

311 

120 

55 

29 

% 

2,530 

.2 

312 

130 

60 

31 

*Me 

2,970 

T3 

313 

140 

65 

33 

% 

3,410 

M 

314 

150 

70 

35 

1Me 

3,895 

" 

315 

160 

75 

37 

1 

4,400 

316 

170 

80 

39 

IKe 

4,995 

317 

180 

85 

41 

IK 

5,500 

318 

190 

90 

43 

IMe 

6,160 

319 

200 

95 

45 

6,820 

320 

215 

100 

47 

IMe 

7,435 

321 

225 

105 

49 

jag 

8,140 

322 

240 

110 

50 

1M 

9,680 

403 

62 

17 

17 

H 

770 

404 

72 

20 

19 

Me 

1,145 

405 

80 

25 

21 

1,385 

406 

90 

30 

23 

11,^  6 

1,650 

407 

100 

35 

25 

s^ 

1,980 

408 

110 

40 

27 

*Me 

2,310 

409 

120 

45 

29 

% 

2,640 

s 

410 

130 

50 

31 

*Me 

3,080 

*n 

411 

140 

55 

33 

1 

3,465 

a 

412 

150 

60 

35 

IMe 

4,400 

413 

160 

65 

37 

4,950 

;> 

414 

180 

70 

42 

1/4 

6,095 

a] 
V 

415 

190 

75 

45 

IMe 

6,710 

•   416 

200 

80 

48 

\% 

7,370 

417 

210 

85 

52 

1K6 

8,030 

418 

225 

90 

54 

9,460 

419 

250 

95 

55 

1  1Me 

10,390 

420 

265 

100 

60 

!1Ke 

12,650 

421 

290 

105 

65 

1  % 

13,500 

422 

320 

110 

70 

2 

15,400 

578 


S.  K.  F.  RADIAL  BEARINGS 


[CHAP.  XX 


by  multiplying  the  capacity  given  in  the  table  by  the  speed 
coefficient  obtained  from  Fig.  335.  The  graph  of  Fig.  335  is 
based  upon  information  published  in  the  trade  literature  issued 
by  the  Hess-Bright  Mfg.  Co. 

(b)  S.  K.  F.  radial  bearing. — The  leading  dimensions  and  load 
rating  of  the  light,  medium,  and  heavy  series  of  the  narrow 
type  of  S.K.F.  self-aligning  radial  bearing,  similar  to  that  shown 
in  Fig.  325(d),  are  given  in  Table  105.  The  load  rating  given  in 


i.o 


0.9 


c    0.8 

_<U 
U 

| 

<D 


fl    0.6 


0.5 


500 
Revolu+ions      per     nfiinu+e 

FIG.  335. 


1000 


1500 


this  table  applies  to  a  steady  load  and  a  constant  speed  not 
exceeding  300  revolutions  per  minute.  To  determine  the  per- 
missible load  capacity  for  a  bearing  running  at  other  speeds  than 
300,  the  rating  given  in  the  table  must  be  multiplied  by  the  speed 
coefficient  obtained  from  the  graph  of  Fig.  336.  This  graph  was 
plotted  from  data  deduced  from  the  load-carrying  capacities  and 
speeds  given  in  the  trade  publication  issued  by  the  S.  K.  F. 
Ball  Bearing  Co. 

(c)  F.  and  S.  thrust  bearing. — The  dimensions  and  load  ratings 
given  in  Table  106  pertain  to  the  light,  medium,  and  heavy  series 
of  F.  and  S.  spherical  seated  type  of  ball  bearing  shown  in  Fig. 


ART.  395]  S.  K.  F.  RADIAL  BEARINGS  579 

TABLE  105. — DATA  PERTAINING  TO  S.  K.  F.  RADIAL  BEARINGS 


No.  and  type 

Dimensions  in  mm. 

Capacity 

of 
bearing 

1 

2 

3 

4 

5 

at 
300  r.p.m. 

1,200 

30 

10 

9 

1 

200 

1,201 

32 

12 

10 

1 

220 

1,202 

35 

15 

11 

1 

285 

1,203 

40 

17 

12 

1 

365 

1,204 

47 

20 

14 

1 

465 

1,205 

52 

25 

15 

1 

630 

1,206 

62 

30 

16 

1 

850 

1,207 

72 

35 

17 

2 

970 

8 

1,208 

80 

40 

18 

2 

1,210 

'C 

1,209 

85 

45 

19 

2 

1,400 

DO 

1,210 

90 

50 

20 

2 

1,575 

^3 

1,211 

100 

55 

21 

2 

1,930 

3 

1,212 

110 

60 

22 

2 

2,205 

h3 

1,213 

120 

65 

23 

2 

2,430 

1,214 

125 

70 

24 

2 

2,810 

1,215 

130 

75 

25 

2 

2,980 

1,216 

140 

80 

26 

3 

3,310 

1,217 

150 

85 

28 

3 

4,080 

1,218 

160 

90 

30 

3 

4,630 

1,219 

170 

95 

32 

3 

5,520 

1,220 

180 

100 

34 

3 

6,060 

1,221       190 

105 

36 

3 

7,060 

1,222       200 

110 

38 

3 

7,720 

1,300 

35 

10 

11 

1 

265 

1,301 

37 

12 

12 

1 

350 

1,302 

42 

15 

13 

1 

385 

1,303 

47 

17 

14 

1 

550 

1,304 

52 

20 

15 

1 

575 

1,305 

62 

25 

17 

1 

885 

1,306 

72 

30 

19 

2 

1,100 

1,307 

80 

35 

21 

2 

1,430 

.8 

1,308 

90 

40 

23 

2 

1,760 

&> 

1,309 

100 

45 

25 

2 

2,200 

1,310 

110 

50 

27 

2 

2,540 

s 

1,311 

120 

55 

29 

2 

3,310 

3 

1,312 

130 

60 

31 

2 

3,860 

1 

1,313 

140 

65 

33 

3 

4,410 

^ 

1,314 

150 

70 

35 

3 

5,070 

1,315 

160 

75 

37 

3 

5,850 

1,316 

170 

80 

39 

3 

6,070 

1,317 

180 

85 

41 

3 

42  '.6 

8,720 

1,318 

190 

90 

43 

3 

44.0 

9,100 

1,319 

200 

95 

45 

3 

46.0 

10,900 

1,320 

215 

100 

47 

3 

58.0 

11,400 

1,321 

225 

105 

49 

3 

60.0 

13,200 

1,322 

240 

110 

50 

3 

64.0 

14,300 

402 

52 

15 

15 

1 

630 

403 

62 

17 

17 

1 

885 

404 

72 

20 

19 

2 

1,145 

405 

80 

25 

21 

2 

1,435 

406 

90 

30 

23 

2 

1,765 

407 

100 

35 

25 

2 

2,205 

408 

110 

40 

27 

2 

2,535 

S 

409 

120 

45 

29 

2 

3,300 

'S 

410 

130 

50 

31 

2 

3,850 

03 

411 

140 

55 

33 

3 

4,410 

£ 

412 

150 

60 

35 

3 

5,070 

S 

413 

160 

65 

37 

3 

6,060 

K 

414 

180 

70 

42 

3 

8,820 

415 

190 

75 

45 

3 

9,080 

416 

200 

80 

48 

3 

11,000 

417 

210 

85 

52 

3 

54!  5 

11,000 

418 

225 

90 

54" 

3 

60.0 

13,250 

419 

250 

95 

55 

3 

66.0 

15,100 

420 

265 

100 

60 

3 

68.5 

16,550 

580 


F.  AND  S.  THRUST  BEARINGS 


[CHAP.  XX 


330(6).  Using  the  load  capacity  as  given  in  the  table  as  a  basis, 
the  permissible  maximum  rating  for  a  bearing  running  at  any 
constant  speed  in  excess  of  50  revolutions  per  minute  may  be 


1.2 


Revolutions    per    minu-i-e 
100  200  300  400  500  1000  1600 


1.0 


0.9 

4- 
_ 
cr  0.6 


-a 

0) 

CL 


0.7 


0.6 


0.5 


0.4 


0.3- 


4000 


3500 


3000 


2500 


2000 


1500 


Revolutions     per     minu+e 
FIG.  336. 


determined  by  multiplying  the  tabular  value  by  the  speed  coef- 
ficient obtained  from  Fig.  337.  The  graph  of  Fig.  337  is  based 
upon  the  load  capacities  given  in  the  trade  literature  issued  by 


ART.  395] 


F.  AND  S.  THRUST  BEARINGS 


581 


TABLE  106. — DATA  PERTAINING  TO  F.  &  S.  SPHERICAL  SEATED  THRUST 

BEARINGS 


No.  and  type 

Dimensions  in  mm. 

Balls 

Capacity 

of 
bearing 

1 

2 

3 

4 

5 

6 

No. 

Diam. 

at 
50  r.p.m. 

AJL  10 

30 

10 

14 

12 

21 

21 

8 

y 

675 

AJL  15 

35 

15 

16 

16 

24 

25 

10 

^4 

850 

AJL  20 

40 

20 

16 

21 

26 

30 

12 

J4 

1,000 

AJL  25 

45 

25 

16 

26 

33 

35 

14 

YA 

1,200 

AJL  30 

53 

30 

18 

32 

38 

40 

16 

% 

1,350 

AJL  35 

62 

35 

21 

37 

44 

50 

16 

/I  6 

2,100 

AJL  40 

64 

40 

21 

42 

49 

50 

17 

2,250 

8 

AJL  45 

73 

45 

25 

47 

55 

60 

16 

% 

3,000 

"C 

AJL  50 

78 

50 

25 

52 

60 

65 

18 

% 

3,400 

£ 

AJL  55 

88 

55 

28 

57 

65 

70 

17 

m 

4,400 

AJL  60 

90 

60 

28 

62 

70 

75 

18 

%6 

4,650 

'S 

AJL  65 

100 

65 

32 

67 

75 

80 

18 

6,100 

kJ 

AJL  70 

103 

70 

32 

72 

80 

85 

19 

% 

6,400 

AJL  75 

110 

75 

32 

77 

85 

90 

20 

% 

6,700 

AJL  80 

115 

80 

35 

82 

90 

95 

21 

^2 

7,100 

AJL  85 

125 

85 

38 

88 

97 

105 

18 

% 

9,500 

AJL  90 

132 

90 

39 

93 

103 

110 

19 

% 

10,000 

AJL  95 

140 

95 

41 

98 

109 

115 

18 

*H6 

11,400 

AJL  100 

148 

100 

42 

103 

118 

120 

19 

% 

12,000 

AJL  105 

155 

105 

43 

110 

130 

130 

18 

H 

13,600 

AJL  110 

160 

110 

43 

115 

135 

135 

19 

H 

14,400 

BJL  25 

52 

25 

19 

26 

40 

40 

13 

&« 

1,700 

BJL  30 

60 

30 

21 

32 

45 

45 

13 

2,450 

BJL  35 

68 

35 

24 

37 

55 

55 

13 

%  6 

3,350 

BJL  40 

76 

40 

27 

42 

60 

60 

13 

4,400 

BJL  45 

85 

45 

30 

47 

65 

65 

13 

5Ke 

5,500 

BJL  50 

92 

50 

33 

52 

75 

75 

13 

6,800 

S 

BJL  55 

100 

55 

35 

57 

80 

80 

13 

21^0 

7,500 

•g 

BJL  60 

106 

60 

37 

62 

85 

85 

13 

% 

8,250 

S 

BJL  65 

112 

65 

38 

67 

90 

90 

14 

% 

9,700 

e' 

BJL  70 

120 

70 

40 

72 

95 

95 

14 

H 

10,600 

3 

BJL  75 

128 

75 

43 

77 

105 

105 

14 

4*6 

12,400 

T3 

BJL  80 

136 

80 

46 

82 

110 

110 

14 

14 

14,400 

O> 

BJL  85 

145 

85 

49 

88 

120 

120 

14 

% 

16,500 

S 

BJL  90 

155 

90 

52 

93 

125 

125 

14 

i 

18,800 

BJL  95 

165 

95 

56 

98 

130 

130 

14 

iKe 

21,300 

BJL  100 

172 

100 

59 

103 

140 

140 

14 

24,000 

BJL  105 

180 

105 

62 

110 

145 

145 

14 

l^ie 

26,500 

BJL  110 

190 

110 

65 

115 

155 

155 

14 

IK 

29,500 

BJL  115 

200 

115 

68 

120 

160 

160 

14 

IKe 

32,500 

BJL  120 

210 

120 

72 

125 

170 

170 

14 

ja/ 

35,500 

CJL  20 

50 

20 

21 

21 

35 

35 

10 

% 

1,900 

CJL  25 

60 

25 

25 

26 

45 

45 

10 

JMa 

3,000 

CJL  30 

73 

30 

30 

32 

50 

50 

10 

4,300 

CJL  35 

80 

35 

33 

37 

60 

60 

10 

% 

5,250 

CJL  40 

90 

40 

38 

42 

65 

65 

10 

2Hz 

7,000 

CJL  45 

100 

45 

42 

47 

75 

75 

10 

l*A* 

8,900 

CJL  50 

110 

50 

47 

52 

80 

80 

10 

*%2 

11,000 

S 

CJL  55 

120 

55 

52 

57 

90 

90 

10 

1 

13,500 

CJL  60 

130 

60 

56 

62 

95 

95 

10 

IHe 

15,200 

« 

CJL  65 

140 

65 

61 

67 

105 

105 

10 

19,000 

1 

CJL  70 
CJL  75 
CJL  80 

150 
160 
170 

70 
75 
80 

65 
70 
74 

72 
77 
82 

110 
120 
125 

110 
120 
125 

10 
10 
10 

l! 

21,000 
25,500 
28,000 

w 

CJL  85 

180 

85 

78 

88 

135 

135 

10 

1H 

30,000 

CJL  90 

190 

90 

83 

93 

140 

140 

10 

IH 

35,500 

CJL  95 

195 

95 

86 

98 

145 

145 

10 

1^6 

38,500 

CJL  100 

205 

100 

89 

103 

155 

155 

10 

m 

41,000 

CJL  115 

220 

115 

93 

120 

170 

170 

11 

1^6 

48,500 

CJL  130 

240 

130 

95 

135 

185 

185 

12 

1XM« 

53,000 

CJL  150 

260 

150 

99 

155 

205 

205 

13 

IK 

61,500 

582 


F.  AND  S.  THRUST  BEARINGS  [CHAP.  XX 


1.0 


0.9 


0.8 


0.7 


4- 
u 
.     06 


Revolutions    per    minute 
0  100  200  300  400  500 


0.5 


(U 

a. 
(A 


0.4 


0.5 


02 


•0.1 


2000 


1500 


1000 


500 


Revolutions   per  minute 
FIG.  337. 


ART.  395]  GURNEY  RADIO-THRUST  BEARINGS  583 

The  Bearings  Co.  of  America,  the  distributors  of  the  F.  and  S. 
bearings. 

(d)  Gurney  radio-thrust  bearing. — The  Gurney  radio-thrust 
bearing  is  manufactured  in  the  following  three  standard  types: 

1.  Type  RT  having  a  thrust  capacity  equivalent  to  100  per 
cent,  of  the  rated  radial  load. 

2.  Type  RT  150  having  a  thrust  capacity  equivalent  to  150  per 
cent,  of  the  rated  radial  load. 

3.  Type  RT  200  having  a  thrust  capacity  equivalent  to  200  per 
cent,  of  the  rated  radial  load. 

In  Table  107  are  given  the  leading  dimensions,  load-carrying 
capacity,  and  speed  rating  for  the  light,  medium,  and  heavy  series 
of  the  RT  type  Gurney  radio-thrust  bearing  similar  to  that  shown 
in  Fig.  333.  It  should  be  understood  that  the  ratings  given  in 
Table  107  are  not  intended  for  all  conditions  of  operation,  but 
that  they  apply  only  to  the  class  of  service  in  which  uniform 
load  and  constant  speed  prevail.  Furthermore,  the  speed  rating 
applies  to  the  type  of  mounting  in  which  the  inner  race  rotates. 
When  the  outer  race  revolves,  the  permissible  speed  should  be 
taken  as  60  per  cent,  of  that  listed  in  the  table.  According  to 
information  furnished  by  the  manufacturer,  the  rated  capacity 
of  the  RT  150  type  is  95  per  cent,  and  that  of  the  RT  200  type  is 
90  per  cent,  of  the  rating  given  in  Table  107.  When  the  speed 
of  the  inner  race  is  above  or  below  the  value  given  in  the  table, 
the  permissible  load  capacity  is  obtained  by  multiplying  the  rated 
load  by  the  so-called  load  factor.  This  factor  depends  upon  the 
speed  coefficient  and  may  be  determined  from  the  graph  of  Fig. 
338.  By  the  term  speed  coefficient  is  meant  the  ratio  of  the  given 
revolutions  per  minute  to  the  rated  revolutions  per  minute  given 
in  the  table. 

To  determine  the  capacity  of  a  Gurney  radio-thrust  bearing 
having  given  the  radial  load  coming  upon  it,  the  following  rule 
used  by  the  Gurney  Ball  Bearing  Co.  is  recommended: 

Rule  I. — "Subtract  the  actual  radial  load  from  the  rated  load 
and  multiply  the  remainder  by  the  thrust  percentage  of  the  bearing.'' 

If  it  is  required  to  determine  the  available  radial  load  capacity 
of  a  radio-thrust  bearing  having  given  the  axial  thrust,  the 
following  rule  should  be  used: 

Rule  II. — "Divide  the  actual  thrust  by  the  thrust  percentage 
of  the  bearing  and  subtract  the  result  from  the  rated  load." 


584 


GURNEY  RADIO-THRUST  BEARINGS        [CHAP.  XX 


The  use  of  the  various  factors  and  rules  just  given  is  shown 
best  by  applying  them  to  a  problem,  as  follows: 


Speed       Coe-Fficien-f 
0.5  1.0  1.5 


2.0 


•3.0' 


3.5 


2.5 


2.0 


1.0 


5.0 


4.0  3.5 

Speed     Coe -P-Fi c i e n  -f 
FIG.  338. 


3.0 


2.5 


Problem. — It  is  required  to  determine  the  thrust  capacity  of  No.  310  RT 
150  radio-thrust  bearing  running  at  750  revolutions  per  minute,  assuming 
that  the  bearing  is  carrying  a  radial  load  of  1,000  pounds. 

Solution.— The  load  rating  of  bearing  No.  310  RT  150  is  3,000  X  0.95  or 
2,850  pounds  at  625  revolutions  per  minute.  From  Fig.  338  the  load  factor 

corresponding  to  a  speed  factor  of  T^TK,  or  1-2,  is  0.91,  hence  the  load  capacity 

at  750  revolutions  per  minute  is  2,850  X  0.91  or  2,590  pounds.  Applying 
the  first  rule  given  above,  the  thrust  capacity  of  the  given  bearing  operating 
under  the  above  conditions  is  1.5  (2,590  —  1,000)  or  2,385  pounds. 


ART.  395]  GURNEY  RADIO-THRUST  BEARINGS  585 

TABLE  107. — DATA  PERTAINING  TO  GURNET  RADIO-THRUST  BEARINGS 


Dimensions 

Balls 

No.  and  type 

Load 

Speed 

of 
bearing 

1 

2 

3 

4 

No. 

Diam., 

rating 

rating 

mm. 

mm. 

mm. 

in. 

in. 

204  RT 

47 

20 

14 

Hi 

13 

%2 

480 

1,450 

205  RT 

52 

25 

15 

/  6  4 

14 

% 

530 

1,250 

206  RT 

62 

30 

16 

7i  6 

15 

"H 

720 

1,050 

207  RT 

72 

35 

17 

H  6 

16 

1,100 

900 

208  RT 

80 

40 

18 

He 

16 

P 

1,480 

800 

209  RT 

85 

45 

19 

He 

17 

H 

1,570 

740 

8 

210  RT 

90 

50 

20 

Hi 

18 

Ji 

1,670 

680 

*c 

211  RT 

100 

55 

21 

Hi 

18 

M 

1  960 

600 

% 

212  RT 

110 

60 

22 

Hi 

17 

5Ke 

2,570 

550 

** 

213  RT 

120 

65 

23 

Hi 

17 

3,230 

500 

60 

214  RT 

125 

70 

24 

H* 

18 

H 

3,420 

475 

3 

215  RT 

130 

75 

25 

19 

H 

3,610 

450 

216  RT 

140 

80 

26 

5^2 

19 

aHe 

4,370 

420 

217  RT 

150 

85 

28 

To  2 

18 

H 

4,850 

390 

218  RT 

160 

90 

30 

18 

4ie 

5,700 

370 

219  RT 

170 

95 

32 

H 

18 

H 

6,650 

350 

220  RT 

180 

100 

34 

l^j 

17 

lMe 

7,220 

330 

221  RT 

190 

105 

36 

y% 

17 

1 

8.080 

315 

222  RT 

200 

110 

38 

y» 

17 

IHe 

9,310 

300 

304  RT 

52 

20 

15 

H* 

13 

I/6 

620 

1.200 

305  RT 

62 

25 

17 

He 

13 

860 

1,050 

306  RT 

72 

30 

19 

He 

13 

He 

1,190 

950 

307  RT 

80 

35 

21 

He 

13 

1,570 

850 

308  RT 

90 

40 

23 

H* 

13 

%  6 

2,000 

750 

309  RT 

100 

45 

25 

Hi 

13 

% 

2,470 

675 

s 

310  RT 

110 

50 

27 

Hi 

13 

% 

3,000 

625 

311  RT 

120 

55 

29 

13 

H 

3,570 

575 

si 

312  RT 

130 

60 

31 

%2 

14 

'He 

4,470 

525 

g 

313  RT 

140 

65 

33 

*Az 

14 

H 

5,230 

475 

3 

314  RT 

150 

70 

35 

14 

^ie 

5,990 

450 

•3 

315  RT 

160 

75 

37 

y» 

14 

1 

6,750 

425 

V 

316  RT 

170 

80 

39 

y& 

14 

IHe 

'7,600 

400 

§ 

317  RT 

180 

85 

41 

L£ 

14 

8,550 

375 

318  RT 

190 

90 

43 

M 

14 

l*!e 

9,590 

350 

319  RT 

200 

95 

45 

/^ 

14 

IK 

10,640 

330 

320  RT 

215 

100 

47 

y» 

14 

l|ie 

11,740 

315 

321  RT 

225 

105 

49 

H 

14 

12,830 

300 

322  RT 

240 

110 

50 

y& 

14 

1>I 

15,200 

285 

404  RT 

72 

20 

19 

H  6 

9 

Ke 

1,380 

1,250 

405  RT 

80 

25 

21 

He 

9 

H 

1,620 

1,075 

406  RT 

90 

30 

23 

Hi 

10 

% 

2,280 

875 

407  RT 

100 

35 

25 

Hi 

10 

% 

2,660 

775 

408  RT 

110 

40 

27 

Hi 

11 

lHe 

3,520 

700 

409  RT 

120 

45 

29 

Hi 

11 

H 

4,090 

650 

410  RT 

130 

50 

31 

H* 

11 

lKe 

4,700 

6uO 

$ 

411  RT 

140 

55 

33 

Ht 

11 

1 

5,320 

550 

S 

412  RT 

150 

60 

35 

Ha 

11 

IHe 

5,990 

500 

oo 

413  RT 

160 

65 

37 

%2 

11 

1^ 

6,750 

460 

>> 

to 

414  RT 

180 

70 

42 

11 

IKe 

9,190 

425 

S 

415  RT 

190 

75 

45 

y& 

11 

IH 

10,070 

400 

W 

416  RT 

200 

80 

48 

y% 

11 

IHe 

11,020 

375 

417  RT 

210 

85 

52 

y& 

11 

1H 

11,970 

350 

418  RT 

225 

90 

54 

y% 

11 

1% 

14,060 

325 

419  RT 

250 

95 

55 

y& 

11 

jax 

16,340 

300 

420  RT 

265 

100 

60 

y% 

11 

1  J^ 

18,810 

285 

421  RT 

290 

105 

65 

y& 

11 

2V^ 

23,130 

270 

422  RT 

320 

110 

70 

y* 

11 

2He 

28,500 

250 

Problem. — It  is  required  to  determine  the  radial  load  capacity  of  bearing 
No.  310  RT  150  running  at  750  revolutions  per  minute  and  carrying  an  axial 
load  of  2,100  pounds. 

Solution. — Applying  Rule  II,  we  find  that  the  axial  load  of  2,100  pounds  is 

2  1 00 

equivalent  to    ',  -    or  1,400  pounds.     According  to  the  preceding  problem 
i.o 


586 


MOUNTING  BALL  BEARINGS 


[CHAP.  XX 


the  load  capacity  of  the  given  size  of  bearing  running  at  750  revolutions  per 
minute  is  2,590  pounds,  whence  the  magnitude  of  the  radial  load  that  may  be 
placed  upon  the  bearing  in  addition  to  the  2,100  pounds  axial  load  is 
2,590  -  1,400  or  1,190  pounds. 

396.  Mounting  Ball  Bearings. — (a)  Radial  bearings. — In 
genera],  the  requirements  of  a  correct  mounting  for  radial 
ball  bearings  carrying  no  axial  thrust  are  as  follows. 

1.  The  shaft  and  sleeves  or  bosses  of  pulleys,  sprockets,  and 
gears  upon  which  the  inner  races  of  radial  bearings  are  to  be 
mounted  must  be  turned  and  ground  accurately,  and  the  hous- 
ings into  which  the  outer  races  are  fitted  must  be  bored  true  so 
as  to  insure  the  concentric  running  of  the  races. 

2.  In  some  installations,  as  for  example  on  a  line-  or  counter- 
shaft, it  is  impossible  to  provide  the  inner  race  with  a  driving 


FIG.  339. 


fit  on  the  shaft  and  to  clamp  it  against  a  shoulder.  In  such 
cases  a  device  called  an  adapter  is  used.  The  adapter  is  nothing 
more  than  a  split  conical  sleeve  fitted  over  the  shaft  and  pro- 
vided with  a  nut  and  nut-lock  as  shown  in  Fig.  339.  It  is  evi- 
dent that  the  inner  race  may  be  rigidly  clamped  to  the  shaft 
at  any  desired  position. 

3.  Whenever  it  is  necessary  to  use  a  split  housing  the  parts 
must  be  fitted  together  correctly  so  that  the  inner  race  will  not 
become  distorted  due  to  any  clamping  action  of  the  housing. 

4.  The  inner  race  of  the  bearing  must  be  retained  in  a  fixed 
position.     To  this  end  it  is  made  with  a  tight  fit  on  the  shaft  and 
is  held  rigidly  in  position  against  a  shoulder  on  the  shaft  by 
means  of  a  nut  and  nut-lock.     In  Figs.  170  and  171  are  shown 
typical  mountings  of  radial  bearings. 


ART.  396]  MOUNTING  BALL  BEARINGS  587 

5.  The  outer  race  should  readily  take  up  a  free  position  rela- 
tive to   the  balls  and  inner  race,  thus  insuring  a  more  nearly 
perfect  distribution  of  the  load  over  the  entire  outer  race.     To 
permit  a  slight  degree  of  axial  movement,  the  outer  race  should 
have  a  so-called    " sucking  fit"  in  the  housing  and  should  never 
be  held  in  place  rigidly. 

6.  The  mounting  must  be  so  designed  that  the  ball  bearing  will 
not  be  left  exposed  to  the  action  of  water,  dust,  grit,  and  other 
foreign  matter.     Provision  must  also  be  made  for  retaining  the 
lubricant.     Various  forms  of  caps  for  closing  the  sides  of  the  hous- 
ing are  used,  some  of  which  are  shown  in  Figs.  170,  194,  195,  and 
339.     Experience  has  demonstrated  that  one  or  two  cored  grooves 
in  the  caps  packed  with  a  stiff  grease  or  felt  are  effective  in  keep- 
ing out  grit  and  water,  and  in  preventing  the  escape  of  the  lubri- 
cant.    The  grooves  are  generally  made  from  %  to  %  inch  deep, 
and  the  bore  through  which  the  shaft  passes  must  be  made  ap- 
proximately Jf-4  inch  larger  than  the  shaft. 

7.  Whenever  a  shaft  having  no  thrust  bearing  is  supported  by 
several  radial  bearings,  satisfactory  results  are  assured  by  secur- 
ing the  outer  race  of  one  of  these  bearings  against  axial  move- 
ment, while  the  outer  races  of  the  remaining  bearings  must  be 
left  free  to  locate  themselves. 

(b)  Single-thrust  bearings. — In  mounting  one-direction  thrust 
bearings  the  rotating  race  must  be  pressed  against  a  suitable 
shoulder  on  the  shaft  by  a  light  driving  fit.  The  shoulder  on  the 
shaft  must  be  of  sufficient  height  so  as  not  to  subject  the  rotating 
race  to  an  undue  bending  action.  If  it  is  impossible  to  provide  a 
proper  shoulder,  a  suitable  washer  or  sleeve  must  be  used  between 
the  shoulder  and  the  race.  The  bore  of  the  stationary  race  is 
made  considerably  larger  than  that  of  the  rotating  race.  To 
secure  satisfactory  service  with  a  thrust  bearing,  the  load  upon 
the  balls  must  be  distributed  evenly.  For  this  purpose  the  sta- 
tionary race  is  provided  with  a  spherical  face,  so  that  the  complete 
thrust  bearing  may  be  supported  on  a  spherical  seated  washer. 
The  bore  of  the  latter  must  be  large  and  in  the  majority  of  instal- 
lations, this  washer  should  be  free  to  move  laterally,  thus  provid- 
ing for  any  shaft  deflection  that  is  liable  to  occur.  Whenever 
possible  the  center  of  the  spherical  seated  race  and  washer 
should  be  located  at,  or  near,  the  center  of  the  radial  bearing  used 
in  conjunction  with  the  thrust  bearing.  As  in  the  case  of  radial 
bearings,  the  mountings  of  all  types  of  thrust  bearings  must  ex- 


588  MOUNTING  BALL  BEARINGS  [CHAP.  XX 

elude  water,  grit,  and  foreign  matter  and  at  the  same  time  retain 
the  lubricant.  Furthermore,  the  same  degree  of  workmanship  on 
the  various  parts  of  the  thrust  bearing  mounting  is  required  as  is 
necessary  for  the  radial  bearing. 

(c)  Double-thrust  bearing. — In  mounting  a  double-thrust  bear- 
ing  of  the  type  illustrated   in   Fig.  331,  the   spherical    faced 
stationary  or  outer  ball  races  must  bear  against  accurately  ma- 
chined spherical  seats  in  the  housing  and  cap.     The  central  or 
rotating  race  must  be  fastened  to  the  shaft  and  held  against  a 
suitable  shoulder  by  means  of  a  sleeve  and  nut-lock.     The  self- 
contained  double-thrust  bearing  shown  in  Fig.  331  is  mounted 
by  fastening  the  central  race  to  the  shaft  in  a  manner  similar  to 
that  described  for  the  plain  double-thrust  bearing.     It  is  unneces- 
sary to  have  an  accurately  bored  seat  for  the  outer  casing,  but 
machined  faces  must  be  provided  between  which  the  entire  bear- 
ing may  be  clamped. 

(d)  Radio-thrust   bearing. — The   single   radio-thrust   bearings 
shown  in  Figs.  325(6)  and  333  can  take  a  thrust  in  only  one  direc- 
tion and  for  that  reason  some  care  must  be  exercised  in  mounting 
such  bearings.     The  Gurney  radio-thrust  bearing  is  made  with 
a  certain  amount  of  looseness  which  must  be  taken  up  in  the 
mounting.     As  in  the  case  of  the  radial  bearings,  the  inner  race 
is  mounted  on  the  shaft  with  a  light  press  fit  and  held  against  a 
suitable  shoulder  on  the  shaft  by  means  of  a  nut  and  nut-lock. 
The  outer  race  is  made  a  push  fit  in  the  housing  and  must  be  held 
against  a  suitable  shoulder  in  the  housing  or  against  the  end  of  an 
adjustable  cap.     Typical  mountings  of  radio-thrust  bearings  are 
shown  in  Figs.  171  and  195. 

The  double-row  bearing  shown  in  Fig.  334  reaUy  consists  of 
two  radio-thrust  bearings  located  within  an  outer  casing.  Since 
this  type  of  bearing  is  capable  of  supporting  a  radial  load  and  at 
the  same  time  take  a  thrust  in  either  direction,  the  method  of 
mounting  depends  upon  the  nature  of  the  loading.  As  in  the 
radial  bearings,  the  inner  race  is  fixed  to  the  shaft,  and  if  axial 
loads  from  both  directions  must  be  taken  care  of,  the  outer  race 
must  be  clamped  rigidly  between  a  suitable  shoulder  on  the 
machine  frame  and  the  end  of  an  adjustable  cap,  as  shown  in 
Fig.  194.  An  installation  of  an  ordinary  radial  bearing  used  in 
conjunction  with  a  double-row  radio-thrust  bearing,  the  latter 
taking  a  thrust  in  one  direction  only,  is  shown  in  Fig.  170. 


ART.  396]  REFERENCES  589 

References 

Bearings  and  their  Lubrication,  by  L.  P.  ALFORD. 
Handbook  for  Machine  Designers  and  Draftsmen,  by  F.  A.  HALSEY. 
Bearings,  Trans.  A.  S.  M.  E.,  vol.  27,  p.  441. 
Ball  Bearings,  Trans.  A.  S.  M.  E.,  vol.  29,  p.  367. 

Ball  Bearings,  Paper  before  Electric  Vehicle  Assoc.  of  Amer.,  Apr.  14,  1915. 
The  Design  of  Ball  Bearings,  Ind.  Eng'g  of  the  Eng'g  Digest,  vol.  13,  pp. 
24,  71,  117. 

A  Series  of  Tests  on  Roller  Bearings,  Amer.  Mach.,  vol.  38,  p.  769 

The  Friction  of  Roller  Bearings,  Mchy.,  vol.  12,  p.  62. 

The  Manufacture  of  Steel  Balls,  Mchy.,  vol.  18,  p.  590. 

Gurney  Ball  Bearing  Engineering  Bulletins,  Gurney  Ball  Bearing  Co. 

Ball  Bearing  Engineering,  Hess-Bright  Mfg.  Co. 

Ball  Bearing  Applications,  The  New  Departure  Mfg.  Co. 

Bulletins  published  by  S.  K.  F.  Ball  Bearing  Co. 


GENERAL  INDEX 


Acme  thread,  79 

Addendum,  281 

Adjustments  for  alignment,  526 

for  wear,  519 
Akron  clutch,  432 
Alco  clutch,  433 
Alignment,  adjustment  for,  526 
Aluminum,  41 

bronze,  41 

copper,  42 
-zinc,  42 

zinc,  42 

Anchors,  chain,  220 
Angle  and  plate  connection,  double, 
68 

single,  64 
Annealing,  43 
Antifriction  curve,  554 
Arms  of  spur  gears,  307 
Auburn  thrust  ball  bearings,  572 
Automatic  brakes,  476 

analysis  of,  484 
Automobile  bolts,  82 
Axial  brakes,  472 

Babbitt  metal,  42,  515 
Bakelite  Micarta-D  gears,  306 
Ball,  load  per,  575 
Balls,  crushing  strength  of,  575 
Ball  bearing,  Auburn  thrust,  472 

Gurney  radio-thrust,  573 

Norma,  570 

Radax,  573 

radial  and  thrust,  573 

S.  K.  R,  569 

Ball  bearings,  coefficient  of  friction 
of,  576 

data  for,  576 

duplex,  574 

mounting  of,  586 

pressures  on,  575 

radial,  569 

thrust,  570 


Ball  raceways,  forms  of,  566 
Band  brakes,  469 

clutch,  454 

analysis  of  a,  456 
Bar,  eccentrically  loaded,  12 

straight  prismatic,  12 
Barth  key,  113 

Beams,  end  connections  for,  66 
Bearing,  Auburn  thrust  ball,  572 

bolts,  design  of,  534 

caps,  design  of,  534 

DeLaval  thrust,  544 

design,  formulas  for,  532 

friction,  coefficient  of,  530 

four-part,  520 

Gurney  radio-thrust  ball,  573 

Hyatt,  559 

materials,  514 

Mossberg  roller,  556 

Norma  ball,  570 
roller,  557 

pintle,  546 

pressures,  527 

on  power  screws,  108 
table  of,  528 

Radax  ball,  573 

radial  roller,  556 

S.  K.  F.  ball,  569 

split,  519 

Timken  roller,  559 

with  thrust  washers,  543 
Bearings,  capacities  of  Norma,  563 

collar  thrust,  543 

conical  radial  roller,  558 
roller  thrust,  561 

connecting  rod,  523 

cylindrical  roller  thrust,  560 

data  for  baU,  576 
for  roller,  562 

dimensions  of  Hyatt,  564 
of  Norma,  563 

duplex  ball,  574 

flexible  roller,  559 


591 


592 


GENERAL  INDEX 


Bearings,  for  friction  gearing,  277 
for  radial  and  axial  loads,  545 
friction  of  collar  thrust,  550 
journal,  513 
length  of,  529 
marine  thrust,  543 
mounting  of  ball,  586 

of  roller,  564 
multiple  disc  step,  547 
pressures  on  ball,  575 
proportions    of   common  split, 

540 

of  journal,  539 
of  pedestal,  540 
of  post,  540 
radial  and  thrust  ball,  573 

ball,  569 

radiating  capacity  of,  530 
right  line,  513 
single  disc  step,  546 
sliding,  513 
solid,  520 
step,  546 

table  of  length  of,  529 
temperature  of,  533 
'      thrust,  513 
ball,  570 
Tower's  experiments  on  collar 

thrust,  551 
on  step,  552 

work  lost  in  collar  thrust,  550 
in  conical  journal,  537 
in  cylindrical  journal,  535 
in  Schiele  pivot,  553 
Becker's  brake,  477 
analysis  of,  477 
Belt,  block  type  of  V,  171 
chain  type  of  V,  172 
fastenings,  tests  of,  155 
ratio  of  tensions,  156 
selection  of  size,  160 
tandem  transmission,  162 
Belts,  tension  in,  155 
Bel-ting,  analysis  of  V,  172 

coefficient  of  friction  for,  159 
experiments  on  steel,  152 
leather,  147 
rubber,  148 
Steel,  151 


Belting,  strength  of  leather,  149 

of  rubber,  150 

Taylor's  experiments  on,  161 
textile,  150 

working  stresses  for  leather,  160 
Bending  moments  on  shafts,  500 

stresses  in  wire  rope,  196 
Bessemer  process,  34 
Bevel-friction  gearing,  266 
Bevel  gear  teeth,  form  of,  323 
Bevel  gears,  acute-angle,  326 
advantages  of  spiral,  345 
arms  of,  339 
bearing  pressures  due  to,  338 

due  to  spiral,  346 
disadvantages  of  spiral,  345 
Fabroil,  341 
mounting,  341 
obtuse-angle,  328 
resultant  pressure  on,  336 
right-angle,  330 
skew,  349 
spiral,  344 
strength  of  cast,  331 

of  cut,  333 
tests  on,  348 
thrusts  due  to,  338 
due  to  spiral,  346 
Birnie's  formula,  131 
Billings  and  Spencer  drop  hammer, 

262 
Block  brakes,  463 

analysis  of,  466 

graphical  analysis  of  double, 

468 
chain,  239 

selection  of,  242 
sprockets  for,  240 
table  of  Diamond,  240 
clutches,  444 

analysis  of,  446 

Bocorselski's  universal  joint,  391 
Boiler  brace,  diagonal,  73 
joint,  analysis  of,  55 

design  of,  59 
Bolts,  automobile,  82 
carriage,  81 
coupling,  81 
design  of  bearing,  534 


GENERAL  INDEX 


593 


Bolts,  machine,  80 

patch,  88 

stay,  89 

Bolts  and  nuts,  U.  S.  Standard,  78 
Brace,  diagonal  boiler,  73 
Brake,  analysis  of  automatic,  484 
of  Becker's,  477 

Becker's,  477 

cam,  483 

case,  480 

coil,  482 

crane  disc,  478 

force  analysis  of  a  disc,  475 

graphical  analysis  of  a  block, 
468 

Luder's,  476 

Niles,  478 

Pawlings     and     Harnischfeger, 

479 
Brakes,  analysis  of  block,  466 

automatic,  476 

axial,  472 

band,  469 

block,  463 

clutch,  423 

conical,  472 

differential  band,  471 

disc,  473 

disposal  of  heat  in,  487 

double-block,  463 

mechanical  load,  476 

post,  463 

simple  band,  469 

single-block,  463 
Brass,  cast,  39 

white,  43 

wrought,  39 
Bronze,  aluminum,  41 

commercial,  40 

manganese,  40 

phosphor,  40 
Bronzes  for  bearings,  515 
Butt  joints,  51 
Buttressed  tooth,  314 

Cadillac  clutch,  412 
Cam  brake,  483 
Cap  screws,  table  of,  84 
Carriage  bolts,  81 


Case  brake,  480 
Case-hardening,  44 
Castellated  nut,  93 
Casting,  chilled,  29 

malleable,  29 
Cast  iron,  26 

for  bearings,  516 
vanadium,  27 
Cementation  process,  35 
Chain  anchors,  220 
block,  239 

analysis  of,  225 
closed  joint,  230 
coil  hoisting,  216 
conveyor,  228 
Conventry  silent,  247 
design  data  for  Morse,  252 
detachable,  228 
drums,  218 
length  of  roller,  246 
Link  Belt  silent,  249 
lubrication  for  bearings,  517 
Morse  silent,  250 

proportions  of  sprockets  for 

conveyor,  238 
proportions    of    sprockets    for 

Link  Belt  silent,  256 
relation     between     effort     and 

load  for,  224 
roller,  242 

selection  of  block,  242 
sheaves,  plain,  221 
pocket  type,  222 
silent,  247 
sprockets  for  detachable,  234 

for  silent,  251 

strength  of  closed  joint,  232 
of  detachable,  229 
of  silent,  250 
stud  link,  217 
table  of,  217 

of  Diamond  block,  240 

roUer,  244 
of  Ewart,  230 
ef  Jeffrey-Mey-Obern,  232 
of  Link  Belt  "400"  class, 

232 

silent,  254 
of  Union  steel,  233 


594 


GENERAL  INDEX 


Chain,  table  of  Whitney  silent,  253 

Whitney  silent,  248 
Chilled  casting,  29 
Chrome  steel,  37 
Chromium-nickel  steel,  37 

-vanadium  steel,  38 
Clamp  coupling,  386 

dimensions  of,  388 
Clark  coupling,  400 
Clavarino's  formula,  132 
Clutch,  Akron,  432 

Alco,  433 

analysis  of  a  band,  456 
of  a  block,  446 
of  a  disc,  434 
of  a  double-cone,  419 
of  the  Hele-Shaw,  440 
of  a  jaw,  402 
of  a  single  cone,  413 
of  a  split-ring,  451 

asbestos  fabric  faced  cone,  416 

band,  454 

block,  444 

brakes,  423 

cone  with  cork  inserts,  417 

design  constants  for  disc,  437 

Dodge  disc,  433 

double  cone,  412 

E.  G.  I.,  426 

engaging  device  for  cone,  421 
mechanisms,  458 

experiments  on  a  cone,  418 

face  angle  of  cone,  417 

Ewart  block,  444 

Farrel  band,  454 

Hele-Shaw,  439 

Horton  roller,  457 

Hunter  block,  445 

hydraulically  operated  disc,  427 

Ideal  multi-cone,  441 

Johnson,  451 

Knox  disc,  423 

leather  faced  cone,  416 

Litchfield  band,  455 

machine  tool  block,  446 
split-ring,  449 

materials  for  friction,  406 

Medart  block,  445 

Metten  disc,  427 


Clutch,    Moore  and  White  disc,  442 
motor  car  cone,  411 
National  cone,  412 
Pathfinder  disc,  434 
Plamondon  disc,  426 
positive,  402 

requirements  of  a  friction,  405 
roller,  457 
single  cone,  408 
single  disc,  423 
split-ring,  449 
Velie  disc,  424 
Wellman-Seaver-Morgan  band, 

455 

Clutches,  study  of  cone,  416 
of  disc,  436 
of  split-ring,  453 
Coefficients  of  friction  for  belting, 

159 

for  friction  gearing,  260 
for  ball  bearings,  576 
for  bearings,  530 
for  square  threads,  107 
of  linear  expansion,  22 
Coil  brake,  482 
Cold-rolled  steel,  36 
Collar  nut,  92 

thrust  bearings,  543 
friction  of,  550 
Tower's  experiments  on,  551 
work  lost  in,  550 
Columns,  eccentric  loading  of,  17 

stresses  in,  15 

Compensating  sprocket,  257 
Compression  combined  with  shear- 
ing, 17 
coupling,  386 

Nicholson,  388 
Compressive  stress,  9 
Cone  clutch,  408 

analysis  of  a  double,  419 

of  a  single,  413 
asbestos  fabric  faced,  416 
Cadillac,  412 
engaging  device  for,  421 
experiments  on,  418 
face  angle  of,  417 
Ideal  multi-cone,  441 
leather  faced,  416 


GENERAL  INDEX 


595 


Cone  clutch,  motor  car,  411 
National,  412 
study  of,  416 
with  cork  inserts,  417 

face  angle,  417 
Conical  brakes,  472 
Connecting  rod  bearings,  523 
Continuous  system  of  rope  trans- 
mission, 181 

Contraction,  forces  due  to,  2 
Copper,  aluminum,  42 

zinc  aluminum,  42 
Cork  inserts,  407 

cone  clutch  with,  417 
Cotter  joint,  analysis  of,  122 
Cotton  rope  transmission,  192 
Coupling  bolts,  81 

clamp,  386 

Clark,  400 

compression,  386 

dimensions  of  clamp,  388 

flange,  383 

Francke,  396 

Hooke's,  390 

Kerr,  400 

leather-laced,  395 
-link,  394 

Nicholson  compression,  388 

Nuttall,  399 

Oldham's,  390 

proportions  of  slip,  431 

roller,  389 

rolling  mill,  401 

slip,  430 

Coventry  silent  chain,  247 
Crane  disc  brakes,  478 

drum  shaft,  501 

drums,  design  of,  208 
Critical  pressure,  527 
Crown  friction  gearing,  267 

double,  275 

efficiency  of,  275 
Crucible  steel,  35 
Crushing  strength  of  balls,  575 
Cut  teeth,  proportions  of,  299 
Cutters  for  cycloidal  teeth,  291 

for  involute  teeth,  287 
Cycloidal  teeth,  action  of,  292 
cutters  for,  291 


Cycloidal  teeth,   Grant's  table  for, 

290 

laying  out,  290 
system  of  gearing,  288 
Cylinder  heads,  cast,  135 

riveted,  135 
Cylinders,  thick,  130 
thin,  129 

Dedendum,  281 

Deformation    due    to    temperature 

change,  21 

DeLaval  thrust  bearing,  544 
Design,  principles  governing,  2 
Diagram,  stress-strain,  3 

of  steel,  4 
Diamond    tooth    form    for     roller 

chain,  244 
Disc  brakes,  473 
crane,  478 

force  analysis  of,  475 
clutch,  analysis  of,  434 
design  constants  for,  437 
hydraulically  operated,  427 
single,  423 
study  of,  436 
Dodge  disc  clutch,  432 
Drop-feed  lubrication,  516 
-hammer,  analysis  of,  262 
Billings  and  Spencer,  262 
Toledo  Machine  and  Tool  Co., 

262 
Drums,  chain,  218 

composite  hoisting,  211 
conical  hoisting,  209 
design  of  crane,  208 
wire  rope,  207 
Duplex  ball  bearings,  574 

Eccentric  loading  of  columns,  17 
Efficiency  of  boiler  joints,  57 

of  crown  friction  gearing,  275 

of  manila  rope  transmission,  190 

of  riveted  connections,  55 

of  spur  gears,  319 

of  square  threads,  104 

of  V-threads,  94 

worm  gearing,  373 
E.  G.  I.  clutch,  426 
Elasticity,  modulus  of,  5 


596 


GENERAL  INDEX 


Elastic  limit,  definition  of,  4 
Electro-galvanizing,  46 
End  connections  for  beams,  66 
Endurance,  safe  stress,  21 
Ewart  chain,  table  of,  230 

clutch,  444 

Expansion,  forces  due  to,  2 
Experimental    conclusions   of    Stri- 
beck,  568 

data  on  hoisting  tackle  179,  204 

results  on  friction  gearing,  259 
Experiments  on  a  cone  clutch,  418 

on  worm  gearing,  381 

Tower's   on  collar  thrust  bear- 
ings, 551 
on  step  bearings,  552 

Fabroil  gears,  305 
Factor  of  safety,  22 

table  of,  23 
Factors,  Lewis  for  stub-teeth,  298 

table  of  Lewis,  297 
F.  and  S.  thrust  bearings,  578 

capacities  of,  581 

dimensions  of,  581 
Farrel  band  clutch,  454 
Fastening  with  eccentric  loading,  101 
Fastenings,  tests  of  belt,  155 
Feather  key,  111 
Flange  coupling,  383 

analysis  of,  384 

marine  type,  386 

proportions  of,  385 
Flat  key,  111 
Flexible  gears,  317 

Nuttall,  318 

Flexure  combined  with  direct  stress, 
11 

stresses  due  to,  10 
Flooded  lubrication,  518 
Forced  lubrication,  518 
Forces,  dead  weight,  1 

due  to  change  of  velocity,  1 

due  to  expansion  and  contrac- 
tion, 2 

frictional,  1 

useful,  1 

Formulas  for  bearing  design,  532 
Francke  coupling,  396 


Frictional  forces,  1 

Friction  clutch,  requirements  of  a, 

405 

coefficient  of  for  ball  bearings, 
576 

for  square  threads,  107 
gearing,  259 

application  of  spur,  261 

bearings  for,  277 

bevel,  266 

coefficients  of  friction  for,  280 

crown,  267 

double  crown,  275 

efficiency  of  crown,  275 

experimental  results  on,  259 

grooved  spur,  264 

plain  spur,  260 
keys,  116 

of  collar  thrust  bearings,  550 
of  conical  journal,  537 
of  cylindrical  journal,  535 
of  feather  keys,  117 
of  pivots,  548 
spindle  press,  271 

pressure  developed  by,  273 

Galvanizing,  electro-,  46 

hot,  46 

Gear,  Ingersoll  slip,  317 
Nuttall  flexible,  318 
Pawlings  and  Harnischf  eger  slip, 

316 
teeth,  form  of  bevel,  323 

proportions  of  helical,  353 
strength  of  double-helical,  354 

of  worm,  370 
strengthening,  311 
Gearing,  application  of  spur  friction, 

261 

bevel  friction,  266 
coefficients  of  friction  for  fric- 
tion, 260 

crown  friction,  267 
cycloidal  system,  288 
double  crown  friction,  275 
efficiency  of  crown  friction,  275 

of  worm,  373 

experimental  results  on  friction, 
259 


GENERAL  INDEX 


597 


Gearing,  friction,  259 

force  analysis  of  worm,  371 
grooved  spur  friction,  264 
Hindley  worm,  365 
involute  system,  284 
load  capacity  of  worm,  369 
materials  for  helical,  357 

for  worm,  366 
plain  spur  friction,  260 
safe  working  stresses  for,  301 
straight  worm,  365 
tooth  forms  for  worm,  367 
Gears,  acute-angle  bevel,  326 

advantages  of  double  helical,  351 
applications  of  double  helical, 

352 
arms  for  helical,  363 

for  spur,  307 
Bakelite  Micarta-D,  306 
bearing  pressures  due  to  bevel, 

338 

due  to  spiral  bevel,  346 
circular  herring  bone,  364 
efficiency  of  spur,  319 
Fabroil,  305 
flexible,  317 
hubs  for  spur,  310 
large  spur,  306 
materials  used  in,  299 
mounting  bevel,  341 

helical,  363 

worm,  377 

obtuse-angle  bevel,  328 
proportions  of  rawhide,  304 
rawhide,  302 

resultant  pressure  on  bevel,  336 
right-angle  bevel,  330 
rims  for  helical,  358 

for  spur,  309 
skew  bevel,  349 
slip,  316 
spiral  bevel,  344 
strength  of  cast  bevel,  331 

of  cast  spur,  294 

of  cut  bevel,  333 

of  cut  spur,  296 

of  Wuest,  356 
tandem  worm,  380 
tests  on  bevel,  348 


Gears,  thrusts  due  to  bevel,  338 
due  to  spiral  bevel,  346 

types  of  helical,  350 

unequal  addendum,  313 
Gib-head  keys,  table  of,  119 
Grease  lubrication,  518 
Gurney  radio-thrust  bearings,  583 

capacities  of,  585 

dimensions  of,  585 

Hangers,  shaft,  526 
Hardening,  43 
Heads,  cast  cylinder,  135 
riveted  cylinder,  135 
Heat  treatments,  S.  A.  E.,  44 
Hele-Shaw  clutch,  439 

analysis  of,  440 
Helical  gear  teeth,  proportions  of,  353 

strength  of,  354 
Helical  gears,  advantages  of  double, 

351 

applications  of  double,  352 
arms  for,  363 
materials  for,  357 
mounting,  363 
rims  for,  358 
types  of,  350 

Hess-Bright  radial  ball  bearings,  576 
capacities  of,  577 
dimensions  of,  577 
Hoisting  chain,  coil,  216 
stud  link,  217 
table  of,  217 
drums,  chain,  218 
composite,  211 
conical,  209 
wire  rope,  207 
sheaves,  wire  rope,  204 
tackle,  analysis,  178 

experimental  data  on,  179,  205 
wire  rope,  202 
Hollow  shafts,  509 
Hook  tooth,  314 
Hooke's  coupling,  390 

law,  3 

Horton  roller  clutch,  457 
Hubs  for  spur  gears,  310 
Hunter  clutch,  445 
Hyatt  roller  bearings,  559 


598 


GENERAL  INDEX 


Hyatt  roller  bearings,  capacities  of, 

564 
dimensions  of,  564 

Ideal  multi-cone  clutch,  441 
Involute  system  of  gearing,  284 
teeth,  action  of,  287 
cutters  for,  287 
Grant's  table  for,  286 
laying  out,  285 
Iron,  cast,  26 
pig,  28 
wrought,  30 

Jaw  clutch,  analysis  of,  402 
Jeffrey-Mey-Obern  chains,  table  of, 

232 

Johnson  clutch,  451 
Joints,  analysis  of  boiler,  55 

butt,  51 

design  of  boiler,  59 

efficiency  of  boiler,  57 

failure  of,  53 

lap,  51 

splice,  70 

structural,  64 
Journals,  friction  of  conical,  537 

friction  of  cylindrical,  535 

stiffness  of,  534 

strength  of,  534 

work  lost  in  conical,  537 

work  lost  in  cylindrical,  535 

Kennedy  keys,  114 
Kerr  coupling,  400 
Key,  Earth,  113 

feather,  111 

flat,  111 

Lewis,  113 

pin,  115 

round,  115 

square,  110 

Woodruff,  111 
Keys,  dimensioning  of,  120 

friction,  116 
of  feather,  117 

Kennedy,  114 

on  flats,  115 

strength  of,  116 


Keys,  table  of  gib-head,  119 

of  round,  115 

of  Woodruff,  112 
Key-seats,  effect  of,  511 
Knox  clutch,  423 

Lap  joints,  51 
Leather  belting,  147 

strength  of,  149 
Lenix  system,  162 
Lewis  factors  for  stub-teeth,  298 
table  of,  297 
key,  113 

Limit  of  proportionality,  3 
Link  Belt  silent  chains,  249 

offset  connecting,  14 

spring  cushioned  sprockets,  257 

sprocket  for,  256 

table  of,  254 
Litchfield  clutch,  455 
Lock  nut,  90 

washer,  table  of,  94 
Lubrication,  chain,  517 

drop-feed,  516 

flooded,  518 

forced,  518 

grease,  518 

provisions  for,  516 

ring,  517 

saturated  pad,  517 

system  of,  516 

wick,  517 
Liider's  brake,  476 

Machine  bolts,  80 
screws,  85 

table  of,  86 
Malleable  casting,  29 
Manganese  bronze,  40 
silicon  steel,  38 
steel  casting,  32 

applications  of,  33 
Manila  hoisting  rope,  175 

stresses  in,  176 
rope,    relation    between    effort 

and  load,  176 
sag  of,  188 
selection  of,  192 


GENERAL  INDEX 


599 


Manila  rope,  transmission,  efficiency 
of,  190 

force  analysis  of,  186 
ratio  of  tensions,  184 
sheave  pressures  for,  188 
sheaves  for,  182 
transmission  rope,  182 
Manufacture  of  shafting,  490 
Marine  thrust  bearings,  543 
Materials  for  friction  clutches,  406 
for  gears,  299 
for  springs,  145 

table  of  physical  properties  of,  24 
Maximum  normal  stress  theory,  495 
shear  theory,  497 
strain  theory,  496 
Mechanical  load  brakes,  476 
Medart  block  clutch,  445 
Merchant     and     Evans     universal 

joint,  392 

Modulus  of  elasticity,  5 
of  resilience,  7 

for  steel,  8 
Monel  metal,  41 
Moore  and  White  clutch,  442 
Morse  chain  design  data,  252 
silent  chain,  250 
spring  cushioned  sprocket,  257 
Mossberg  roller  bearing,  556 
Mounting  ball  bearings,  586 

roller  bearings,  564 
Multiple  system  of  rope  transmis- 
sion, 180 

National  clutch,  411 
Nickel  steel,  36 

-chromium  steel,  37 
Niles  brake,  478 
Non-burn,  406 
Norma  ball  bearings,  570 

bearings,  capacities  of,  563 
dimensions  of,  563 

roller  bearing,  557 
Nuts,  castellated,  93 

collar,  92 

lock,  90 

split,  93 

U.  S.  Standard  bolts  and,  78 
Nuttall  coupling,  399 


Oil  grooves,  516 
Oldham's  coupling,  390 
Open-hearth  process,  34 

Pathfinder  clutch,  434 

Patch  bolts,  88 

Pedestal   bearings,    proportions    of, 

540 

Phosphor  bronze,  40 
Physical    properties    of    materials, 

table  of,  24 
Pig  iron,  28 

general  specifications  of,  29 
Pin  key,  115 

plates,  73 
Pins,  table  of  taper,  115 

taper,  124 
Pintle  bearing,  546 
Pipe  thread,  standard,  77 
Pitch,  chordal,  281 

circular,  280 

diametral,  280 
Pivots,  analysis  of  flat,  551 

friction  of,  548 

Schiele,  553 

work  lost  in,  548 
Plamondon  clutch,  426 
Plate  and  double  angle  connection, 
68 

and  single  angle  connection,  64 

thickness  for  boiler  joints,  58 
Plates,  circular,  134 

elliptical,  135 

pin,  73 

rectangular,  132 

square,  133 
Poisson's  ratio,  6 
Post  bearings,  proportions  of,  540 

brake,  463 
Press,  friction  spindle,  271 

pressure  developed  by  friction 

spindle,  273 

Pressures,    allowable   on  ball  bear- 
ings, 575 

bearing,  527 

critical,  527 

table  of  bearing,  528 
Principles  governing  design,  2 
Proportions  of  cast  teeth,  295 


600 


GENERAL  INDEX 


Proportions  of  common  split  bear- 
ings, 540 

of  journal  bearings,  539 

of  pedestal  bearings,  540 

of  post  bearings,  540 
Pulleys,  cork  insert,  165 

cast  iron,  164 

paper,  165 

proportions  of,  167 

steel,  165 

tension,  162 

tight  and  loose,  169 

transmitting  capacity  of,  166 

wood,  165 

Raceway  having  four-point  contact, 

568 

three-point  contact,  567 
two-point  contact,  566 
Raceways,  forms  of  ball,  566 
Radax  ball  bearing,  573 
Radial  and  thrust  ball  bearings,  573 
ball  bearings,  569 
bearings  having  conical  rollers, 

558 

having  cylindrical  rollers,  556 
having  flexible  rollers,  559 
Radiating  capacity  of  bearings,  530 
Rawhide  gears,  302 

proportions  of,  304 
Raybestos,  406 
Relation  between  driving  and  driven 

sprockets,  235 

effort  and  load  for  chain,  224 
for  manila  rope,  176 
for  wire  rope,  195 
Renold  tooth  form,  245 
Repeating  rivet  group,  55 
Resilience,  6 

modulus  of,  7 
for  steel,  8 

Rim  of  spur  gears,  309 
Ring  lubrication,  517 
Ritter's  formula,  15 
Rivet  heads,  forms  of,  50 
holes,  48 

recommended  sizes,  59 
margin,  54 
spacing  for  structural  work,  63 


Rivets,  48 

forms  of,  49 
Rod  ends,  125 
closed,  525 
open,  524 
table  of  B.  and  S.,  128 

of  S.  A.  E.,  127 
Roller  bearings,  conical  radial,  558 

data  for,  562 

flexible,  559 

Mossberg,  556 

mounting  of,  564 

Norma,  557 

radial,  556 

thrust,  560 

Timken,  559 
chain,  242 

Diamond  tooth  form  for,  244 

length  of,  246 

Renold  tooth  form  for,  245 

table  of  Diamond,  244 

sprockets,  243 
clutch,  457 
coupling,  389 
Rolling  mill  coupling,  401 
Rope,  bending  stresses  in  wire,  196 
drums  for  wire  hoisting,  207 
flat  wire,  211 
manila  hoisting,  175 

transmission,  182 
relation  between  effort  and  load 
for  manila,  176 

between  effort  and  load  for 

wire,  195 
sag  of  manila,  188 

of  wire,  214 
selection  of  manila,  192 

of  wire  hoisting,  202 
sheaves  for  wire  hoisting,  204 
stresses  due  to  slack  in  wire,  200 

in  manila  hoisting,  176 
table  of  wire  transmission,  203 

of  strengths  of  wire,  203 
transmission,    continuous    sys- 
tem, 181 

cotton,  192 

multiple  system,  180 

ratio  of  tensions  in  manila,  184 
of  tensions  in  wire,  213 


GENERAL  INDEX 


601 


Rope  transmission,  sheaves  for  wire, 

213 
single   loop  system   of  wire, 

212 

wire  transmission,  212 
Round  keys,  115 
table  of,  115 
Rubber  belting,  148 

S.  A.  E.  heat  treatments,  44 
Sag  of  manila  rope,  188 

of  wire  rope,  214 
Saturated-pad  lubrication,  517 
Schiele  pivot,  553 
Screws,  bearing  pressures  on  power, 

108 

holding  power  of  set,  87 
machine,  85 
set,  85 
table  of  cap,  84 

of  machine,  86 
Sellers  standard  thread,  77 
Semi-steel,  30 
Set  screws,  85 

holding  power  of,  87 
Shaft,  crane  drum,  501 

supporting  one  normal  and  one 

inclined  load,  506 
three  loads,  508 

two  normal  loads  between  bear- 
ings, 502 
with  one  bearing  between 

the  loads,  505 
Shafting,  489 

cold-rolled,  490 
commercial  sizes  of,  490 
design  constants  for,  492,  494 
drawn,  490 

effect  of  key-seats  on,  511 
manufacture  of,  490 
materials  for,  489 
simple  bending  of,  491 

twisting  of,  492 
subjected  to  combined  twisting 

and  bending,  495 
compression,  499 
torsional  stiffness  of,  495 

strength  of,  493 
transverse  stiffness  of,  492 


Shafting,  transverse  strength  of,  491 

turned,  490 
Shafts,  bending  moments  on,  500 

hollow,  509 
Shaw  brake,  481 

analysis  of,  484 

Shearing   combined   with   compres- 
sion, 17 
stress,  9 
tension,  17 
Sheave   pressures   for  manila   rope 

transmission,  188 
Sheaves,  manila  hoisting  rope,  175 

transmission  rope,  182 
plain  chain,  221 
pocket  chain,  222 
wire  hoisting  rope,  204 

transmission  rope,  213 
Shererdizing,  46 
Shrouding,  311 
Silicon-manganese  steel,  38 
S.  K.  F.  ball  bearing,  569 
bearings,  578 
capacities  of,  579 
dimensions  of,  579 
Slip  coupling,  430 

proportions  of,  431 
gears,  316 
Ingersoll,  317 
Pawlings  and  Harnischfeger, 

316 

Splice  joint,  70 
Splines,  integral  shaft,  120 

proportions  of  shaft,  121 
Split  bearing,  519 

nut,  93 

Split-ring  clutches,  449 
analysis  of,  451 
study  of,  453 
Spring  wire  lock,  93 
Springs,  concentric  helical,  139 
conical,  141 
full  elliptic,  145 
helical,  136 
leaf,  142 

materials  for,  145 
semi-elliptic,  143 
torsion,  140 
Sprocket  teeth  factors,  236 


602 


GENERAL  INDEX 


Sprocket  tooth  form  for  conveyor 

chains,  236 

Sprockets,  armor  clad,  234 
block  chain,  240 
compensating,  257 
detachable  chain,  234 
Link     Belt    spring    cushioned, 

257 

Morse  spring  cushioned,  257 
proportions  for  conveyor  chain, 

238 
proportions  of  Link  Belt  silent 

chain,  256 
relation    between    driving    and 

driven,  235 
roller  chain,  243 
silent  chain,  251 
spring  cushioned,  256 
Spur  friction  gearing,  260 
applications  of,  261 
grooved,  264 
gears,  arms  of,  307 
efficiency  of,  319 
hubs  of,  310 
large,  306 
rim  of,  309 
strength  of  cast,  294 

of  cut,  296 
Square  thread,  77 

coefficient  of  friction  for,  107 
Stay  bolts,  89 
Steel  belting,  151 
casting,  31 

manganese,  32 
chrome,  37 

chromium  vanadium,  38 
cold-rolled,  36 
crucible,  35 

modulus  of  resilience  for,  8 
nickel,  36 

-chromium,  37 
semi-  30 

silicon-manganese,  38 
stress-strain  diagram  of,  4 
tungsten,  38 
vanadium,  37 
Step  bearings,  546 
multiple  disc,  547 
single  disc,  546 


Step  bearings,  Tower's  experiments 

on,  552 

Stiffness  of  journals,  534 
Straight  line  formula  for  columns,  16 
Strain,  definition  of,  3 
Strength  of  journals,  534 

ultimate,  5 
Stress,  compressive,  9 

definition  of,  3 

safe  endurance,  21 

safe  working  in  gearing,  301 

shearing,  9 

tensile,  8 

torsional,  10 

Stresses,  allowable  for  boiler  joints, 
57 

direct  combined  with  flexure,  1 1 

due  to  flexure,  10 

suddenly  applied  forces,  18 
temperature  change,  22 

for  leather  belting,  160 

in  bolts  and  screws,  96 

in  columns,  15 

in  manila  hoisting  ropes,  176 

in  wire  rope  due  to  slack,  200 

repeated  low,  20 
high,  19 

working,  22 

-strain  diagram,  3 
of  soft  steel,  4 
Stribeck,    experimental    conclusions 

of,  568 
Stub  teeth,  312 

dimensions  of  Fellows,  313 

Lewis  factors  for,  298 
Studs,  88 

Tackle,  analysis  of  hoisting,  178 

experimental  data  on  hoisting, 

179,  205 

reefed  with  wire  rope,  202 
Taylor's  experiments  on  belting,  161 
Teeth,  dimensions  of  Fellows  stub, 

313 
for  Hindley  worm  gearing,  369 

worm  gearing,  367 
form  of  bevel  gear,  323 
Lewis  factors  for  stub,  298 
proportions  of  cast,  295 


GENERAL  INDEX 


603 


Teeth,  proportions  of  cut,  299 
of  helical  gear,  353 
of  worm  gear,  368 
short,  311 
strength  of  double-helical  gear, 

354 

of  worm  gear,  370 
strengthening  gear,  311 
stub,  312 

Temperature  of  bearings,  533 
Tempering,  43 
Tensile  stress,  8 

Tension  combined  with  shearing,  17 
Theory,    maximum    normal    stress, 

495 

shear,  497 
strain,  496 
Thermoid,  406 
Thread,  Acme,  79 

efficiency  of  square,  104 

of  V,  94 
forms  of,  76 
Sellers  standard,  77 
square,  77 
standard  pipe,  77 
trapezoidal,  79 
Thrust  ball  bearings,  570 

Auburn,  572 
bearings,  collar,  543 
DeLaval,  544 
having  conical  rollers,  561 
having  cylindrical  rollers,  560 
marine,  543 

Timken  roller  bearing,  559 
Toledo  Machine  and  Tool  Co.  drop 

hammer,  262 
Tooth  curves,  282 

forms,  Diamond  sprocket,  244 
conveyor  chain  sprocket,  236 
Renold,  245 
laying  out  cycloid  al,  290 

involute,  285 
Torsional  stress,  10 
Tower's  experiments  on  collar  thrust 

bearings,  551 
on  step  bearings,  552 
Tractrix,  554 
Trapezoidal  thread,  79 
Tungsten  steel,  38 


Union  steel  chain,  table  of,  233 

Universal  joint,  390 
Bocorselski's,  391 
Merchant  and  Evans,  392 

Ultimate  strength,  5 

Vanadium  cast  iron,  27 

chromium-steel,  38 

steel,  37 

V  belt,  block  type,  171 
chain  type,  172 

belting,  analysis  of,  172 
Vilie  clutch,  424 

Washers,  table  of  lock,  94 
Wear,  adjustments  for,  519 
Wellman-Seavers-Morgan    clutch, 

455 
Whitney  silent  chain,  248 

table  of,  253 
Wick  lubrication,  517 
Wire  hoisting  rope,  selection  of,  202 
rope,  bending  stresses  in,  196 
drums  for,  207 
flat,  211 
relation   between   effort   and 

load  for,  195 
sag  of,  214 

sheaves  for  hoisting,  204 
stresses  due  to  slack,  200 
table  of  strengths  of,  203 
transmission,    ratio    of    ten- 
sions, 213 
sheave  for,  213 
single  loop  system,  212 
Woodruff  keys,  111 

table  of,  112 
Work  lost  in  collar  thrust  bearings, 

550 

conical  journals,  537 
cylindrical  journals,  535 
pivot,  548 
Worm,  construction  of,  375 

gear  shaft,  pressures  on,  373 
gearing,  efficiency  of,  373 
experiments  on,  381 
force  analysis  of,  371 
Hindley,  365 
load  capacity  of,  369 


604  GENERAL  INDEX 

Worm  gearing,  materials  for,  366  Worm  shaft,  bearing  pressures   on, 
straight,  365  373 

strength  of  teeth  for,  370  Wrought  iron,  30 

teeth  for  Hindley,  369  Wuest  gears,  strength  of,  356 
tooth  forms  for,  367 

gears,  construction  of,  375  Jl,     poln  *' * 

mounting  of,  377  Yoke  ends   125 

proportions  of,  375  ^S  A  E    127' 

tandem,  380  )f  b>  At  E>'  127 

Sellers,  377  Zinc,  aluminum,  42 


INDEX  TO  AUTHORS,  INVESTIGATORS, 

PUBLICATIONS,  AND 

MANUFACTURERS 


Ahara,  E.  H.,  191 

Akron  Gear  and  Eng'g  Co.,  The,  441 

Albany    Hardware    Specialty    Mfg. 

Co.,  275 

Alford,  L.  P.,  529 
Allis-Chalmers  Co.,  183,  184 
American  Bridge  Co.,  180 
American  Hoist  and  Derick  Co.,  204 
American  Locomotive  Co.,  433 
American  Machinist,  87,   168,  310, 
355,  369,  415,  488,  532,  576 
American  Mfg.  Co.,  192 
American  Pulley  Co.,  The,  165 
American  Society  of  Civil  Engineers, 

16,  180,  188 

American  Society  of  Mechanical 
Engineers,  58,  60,  85,  89, 
90,  106,  161,  192,  259, 
260,  356 

American  Society  of  Testing  Mate- 
rials, 19 

American  Steel  and  Wire  Co.,  196 
American  Tool  Works,  The,  451 
Association  of  Master  Steam  Boiler 

Makers,  55 
Auburn  Ball  Bearing  Co.,  572 

Bach,  C.,  132,  135,  266,  369,  370,  381 

Barr,  J.  H.,  145,  146 

Barth,  C.  G.,  113,  159,  169,  301 

Bartlett,  G.  M.,  244 

Bates,  W.  C.,  354,  355,  356,  363 

Baush  Machine  Tool  Co.,  391 

Bearings  Company  of  America,  The, 

583 

Benjamin,  C.  H.,  132,  168 
Billings  and  Spencer  Co.,  127,  262 
Birnie,  131 

Bonte,  Prof.  H.,  418,  419 
Brown  and  Sharpe   Mfg.  Co.,  287, 

291,  299,  367,  368 


Bruce  Macbeth  Engine  Co.,  The,  394 
Bryson,  132 

Carpenter,  Prof.  R.  C.,  99 
Case  Crane  Co.,  480 
Champion  Rivet  Co.,  50 
Clavarino,  132 

Clyde  Iron  Works  Co.,  208,  412 
Cornell  University,  531 
Wm.    Cramp    and    Sons   Ship  and 
Engine  Bldg.  Co.,  367,  428 

Day,  P.  C.,  353,  356 

DeLaval  Steam  Turbine  Co.,  544 

Diamond  Chain  and  Mfg.  Co.,  240, 

242,  243,  244,  246 
Diamond  Rubber  Co.,  The,  150 
Dodge  Mfg.  Co.,  183,  184,  190 
Douglas,  E.  R.,  488 

Edgar,  John,  415 

Eloesser,  D.,  152 

Eloesser  Steel  Belt  Co.,  151 

Engineers  Club  of  Philadelphia,  296 

Engineering  Magazine,  148,  154 

Falk  Co.,  The,  353,  356,  358,  359, 

363 

Fawcus  Machine  Co.,  354,  355 
Fellows,  E.  R.  299 
Fellows'  Gear  Shaper  Co.,  313 

General  Electric  Co.,  153,  301,  305 
394,  395,  518,  529,  530 

Gleason  Works,  The,  314,  345,  346, 
348 

Goodenough,  G.  A.,  218 

Goss,  W.  F.  M.,  259,  260 

Grant,  G.  B.,  285,  286,  288,  290 

Grant,  R.  H.,  575,  576 

Grant  Gear  Works,  285 

Grashof,  132,  135 


605 


606 


INDEX   TO  AUTHORS 


Graton  and  Knight  Mfg.  Co.,  172 
Greaves  Klusman  Tool  Co.,  451 
Griffin,  C.  L.,  501 
Guest,  Prof.  497 

Gurney  Ball  Bearing  Co.,  343,  573, 
583,  584,  585 

Halsey,  F.  A.,  238 

Hancock,  Prof.,  499 

Hele-Shaw,  Prof.,  439,  440 

Hess-Bright  Mfg.  Co.,  576,  577,  578 

Hewitt,  W.,  214 

Hindley,    365,  366,  369,  376 

Hooke,  3,  201 

Hunt,  C.  W.,  295,  299,  311 

Hyatt  Roller  Bearing  Co.,  559,  564 

Illinois  Steel  Co.,  430 
Illmer,  Louis,  530,  531,  532 
Ingersoll  Milling  Machine  Co.,  317, 

521 
Institution  of  Mechanical  Engineers, 

551 

Jeffrey  Mfg.  Co.,  232,  236 
Johns-Manville  Co.,  406 
Johnson,  Herman,  310 
Johnson,  Thos.  H.,  16 

Kelso,  531 
Kenyon,  E.,  193 
Kerr,  C.  V.,  400 

Keystone-Hindley  Gear  Co.,  369 
Kimball,  D.  C.,  145,  146 
Kingsbury,  Prof.  A.,  106,  107 
Knight,  William,  532 

Lame,  130 

Lanchester,  369,  376 

Lasche,  530,  531 

Lewis,  Wilford,  113,  296,  297,  298, 

301,  311,  333,  355 

Link  Belt  Co.,  229,  232,  237,  249,  251 
Litchfield    Foundry    and    Machine 

Co.,  455 

Logue,  C.  H.,  299,  313 
Lucas  Machine  Tool  Co.,  410 

Machinery,  348 
Marx,  Prof.  G.  H.,  527 


Maurer,  Prof.,  531 

Mechanical    Engineers'    Handbook, 

138,  141 

Merchant  and  Evans  Co.,  392 
Merriman,  Mansfield,  16,  17,  132 
Mesta  Machine  Co.,  306 
Metten,  J.  F.,  428 
Miller,  Spencer,  187 
Mitchell,  S.  P.,  180 
Moore,  H.  F.,  19,  21,  511,  527 
Moore,  L.  E.,  218 
Morrison,  C.  J.,  148,  154,  160 
Morrison,  E.  R.,  143 
Morse  Chain  Co.,  250,  251,  252,  257 

National  Association  of  Cotton  Mfr., 

166 
New  Departure  Mfg.  Co.,  The,  343, 

379,  573,  574 
New    Process    Rawhide    Co.,    The, 

300,  304 

Niagara  Falls  Power  Co.,  561 
Nichols,  Prof.,  488 
Nicholson  and  Co.,  W.  H.,  389 
Niles-Bement-Pond  Co.,  478,  482 
Nordberg  Mfg.  Co.,  115,  464 
Norma  Company  of  America,  The, 

557,  562,  570 
Nuttall  Co.,  The  R.  D..  317,  364,  399 

Pawlings    and    Harnischfeger    Co., 

315,  479,  547 

Pederson,  Axel  K.,  518,  530 
Philosophical  Magazine,  497 
Pinkney,  B.  H.  D.,  87 
Poisson,  6,  497 
Power,  530 

Rankine,  Prof.,  495 
Renold,  Hans,  245,  249 
Ritter,  15,  17 
Roser,  E.,  369,  370,  381 

Saint  Venant,  496 

Sawdon,  W.  M.,  166 

Seeley,  F.  B.,  19,  21 

Sellers  Co.,  William,  377 

Shaw  Electric  Crane  Co.,  481 

S.  K.  F.  Bearing  Co.,  569,  578,  579 


INDEX  TO  AUTHORS 


607 


Smith,  A.  W.,  527 

Smith,  C.  A.  M.,  499 

Smith,  L.  G.,  299 

Society   of   Automobile    Engineers, 

41,  42,  43,  44,  82,  93,  121, 

127,  417,  576 
South  Wales  Institute  of  Engineers, 

193 

Standard  Machinery  Co.,  557 
Stephens- Adamson  Mfg.  Co.,  540 
Stribeck,  Prof.;  381,  527,  562,  568, 
75,  576 

Taylor,  F.  W.,  161 

Thomas,  C.,  531 

Timken  Roller  Bearing  Co.,  379,  559 

Toledo  Machine  and  Tool  Co.,  262 


Tower,  Beauchamp,  531,  551,  552 
Trenton  Iron  Co.,  214 

Union  Chain  and  Mfg.  Co.,  233,  235 
University    of    Illinois    Experiment 

Station,  218,  511 
University  of  Missouri,  532 

Weisbach,  318 

Wellman-Seavers-Morgan  Co.,  455 
Westcott,  A.  L.,  532 
Westinghouse  Electric  and  Mfg.  Co., 

301,  386 
Whitney  Mfg.  Co.,  Ill,  248,  253 

Zeh  and  Hahnemann  Co.,  271 
Zeitschrift     de^    Vereins    deutcher 
Ingenieure,  369,  418 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below, 
or  on  the  date  to  which  renewed.  Renewals  only: 

Tel.  No.  642-3405 

Renewals  may  be  made  4  days  prior  to  date  due. 
Renewed  books  are  subject  to  immediate  recall. 


AUG80 


L-D2lA-10m-8,'73 
(R1902S10)476 — A-31 


General  Library 

University  of  California 

Berkeley 


C   12693 


-, 
UNIVERSITY  OF  CALIFORNIA  LIBRARY 


